Apparatus and method for transmitting and receiving a quasi-cyclic low density parity check code in a multimedia communication system

Info

Patent number: 8918697
Type: Grant
Filed: Nov 12, 2012
Date of Patent: Dec 23, 2014
Patent Publication Number: 20130124938
Assignee: Samsung Electronics Co., Ltd.
Inventors: Hyun-Koo Yang (Seoul), Sung-Hee Hwang (Gyeonggi-do), Seho Myung (Seoul)
Primary Examiner: Christine Tu
Application Number: 13/674,651

Abstract

An apparatus and method are provided for transmitting and receiving a quasi-cyclic Low Density Parity Check (LDPC) code in a multimedia communication system. In the method, a signal transmission apparatus generates a quasi-cyclic LDPC code, and transmits the quasi-cyclic LDPC code to a signal reception apparatus. The quasi-cyclic LDPC code is generated by encoding an information word vector using a child parity check matrix, generated by performing one of a scaling operation, a row separation operation, and a row merge operation on a parent parity check matrix. In the scaling operation, a size of the child parity check matrix is determined. In the row separation operation, each of rows included in the parent parity check matrix is separated. In the row merge operation, the rows included in the parent parity check matrix are merged.

Description

Description

PRIORITY

This application claims priority under 35 U.S.C. §119(a) to Korean Patent Application Serial No. 10-2011-0117855, which was filed in the Korean Intellectual Property Office on Nov. 11, 2011, and to Korean Patent Application Serial No. 10-2012-0111459, which was filed in the Korean Intellectual Property Office on Oct. 8, 2012, the entire disclosure of each of which is hereby incorporated by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to an apparatus and method for transmitting and receiving a quasi-cyclic Low Density Parity Check (LDPC) code in a multimedia communication system, and more particularly to a quasi-cyclic LDPC code transmission and reception apparatus and method that support various codeword lengths and code rates.

2. Description of the Related Art

A multimedia communication system, for example, a Moving Picture Experts Group (MPEG) Media Transport (MMT) system provides various large-capacity content, such as High Definition (HD) content, Ultra High Definition (UHD) content, etc. Further, according to a diversification of this content and especially increases in large-capacity content, data congestion has become a more serious issue. As a result of the data congestion, content transmitted by a signal transmission device is not always completely transferred to a signal reception device, and some of the content is lost en route.

In general, data is transmitted on a packet basis, and accordingly, data loss is generated on a transmission packet basis. For example, if the transmission packet is lost on a network, the signal reception device cannot receive the lost transmission packet, and thus, cannot know the data within the lost transmission packet. As a result, user inconvenience occurs, such as audio quality deterioration, video quality deterioration, video break, caption omissions, file loss, etc.

Therefore, the MMT system may enhance system reliability using various error-control schemes in order to reduce information data loss that is often caused by an error that occurs on a network according to channel status. A typical example of an error-control scheme is an Application Layer-Forward Error Correction (AL-FEC) scheme.

However, a multimedia communication system, such as an MMT system, requires usage of a Forward Error Correction (FEC) code supporting a codeword length and a code rate, which vary according to a code rate and delay time required by a service.

Further, when a conventional AL-FEC scheme is used, a signal transmission and reception apparatus should use different FEC codes according to a codeword length and a code rate, thereby increasing the complexity of the MMT system.

Consequently, it is difficult to implement an MMT system due to this increased complexity.

SUMMARY OF THE INVENTION

Accordingly, the present invention is designed to address at least the problems and/or disadvantages described above and to provide at least the advantages described below.

An aspect of the present invention is to provide an apparatus and method for transmitting and receiving a quasi-cyclic LDPC code in a multimedia communication system.

Another aspect of the present invention is to provide a quasi-cyclic LDPC code transmission and reception apparatus and method that support various codeword lengths in a multimedia communication system.

Another aspect of the present invention is to provide a quasi-cyclic LDPC code transmission and reception apparatus and method that support various code rates in a multimedia communication system.

Another aspect of the present invention is to provide a quasi-cyclic LDPC code transmission and reception apparatus and method that support various codeword lengths using a scaling scheme and a shortening scheme in a multimedia communication system.

Another aspect of the present invention is to provide a quasi-cyclic LDPC code transmission/reception apparatus and method that support various code rates using one of a row separation scheme and a row merge scheme, and a puncturing scheme in a multimedia communication system.

In accordance with an aspect of the present invention, a signal transmission apparatus in a multimedia communication system is provided. The signal transmission apparatus includes a quasi-cyclic Low Density Parity Check (LDPC) code generator for generating a quasi-cyclic LDPC code; and a transmitter for transmitting the quasi-cyclic LDPC code to a signal reception apparatus, wherein the quasi-cyclic LDPC code is generated by encoding an information word vector using a child parity check matrix which is generated by performing one of a scaling operation, a row separation operation and a row merge operation on a parent parity check matrix, and wherein the scaling operation is an operation in which a size of the child parity check matrix is determined, the row separation operation is an operation in which each of rows included in the parent parity check matrix is separated, and the row merge operation is an operation in which the rows included in the parent parity check matrix are merged.

In accordance with another aspect of the present invention, a signal reception apparatus in a multimedia communication system is provided. The signal reception apparatus includes a receiver for receiving a quasi-cyclic Low Density Parity Check (LDPC) code; and a LDPC code decoder for recovering an information word vector by decoding the quasi-cyclic LDPC code, wherein the quasi-cyclic LDPC code is generated by encoding an information word vector using a child parity check matrix which is generated by performing one of a scaling operation, a row separation operation and a row merge operation on a parent parity check matrix, and wherein the scaling operation is an operation in which a size of the child parity check matrix is determined, the row separation operation is an operation in which each of rows included in the parent parity check matrix is separated, and the row merge operation is an operation in which the rows included in the parent parity check matrix are merged.

In accordance with another aspect of the present invention, a method is provided for transmitting a quasi-cyclic Low Density Parity Check (LDPC) code by a signal transmission apparatus in a multimedia communication system. The method includes generating a quasi-cyclic LDPC code; and transmitting the quasi-cyclic LDPC code to a signal reception apparatus, wherein the quasi-cyclic LDPC code is generated by encoding an information word vector using a child parity check matrix which is generated by performing one of a scaling operation, a row separation operation and a row merge operation on a parent parity check matrix, and wherein the scaling operation is an operation in which a size of the child parity check matrix is determined, the row separation operation is an operation in which each of rows included in the parent parity check matrix is separated, and the row merge operation is an operation in which the rows included in the parent parity check matrix are merged.

In accordance with another aspect of the present invention, a method is provided for receiving a quasi-cyclic Low Density Parity Check (LDPC) code by a signal reception apparatus in a multimedia communication system. The method includes receiving a quasi-cyclic LDPC code; and recovering an information word vector by decoding the quasi-cyclic LDPC code, wherein the quasi-cyclic LDPC code is generated by encoding an information word vector using a child parity check matrix which is generated by performing one of a scaling operation, a row separation operation and a row merge operation on a parent parity check matrix, and wherein the scaling operation is an operation in which a size of the child parity check matrix is determined, the row separation operation is an operation in which each of rows included in the parent parity check matrix is separated, and the row merge operation is an operation in which the rows included in the parent parity check matrix are merged.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other aspects, features, and advantages of certain embodiments of the present invention will be more apparent from the following detailed description taken in conjunction with the accompanying drawings, in which:

FIG. 1 is a block diagram illustrating a quasi-cyclic LDPC code generator included in a signal transmission apparatus in an MMT system according to an embodiment of the present invention;

FIG. 2 is a block diagram illustrating a parity check matrix generator according to an embodiment of the present invention;

FIG. 3 schematically illustrates a row separation process according to an embodiment of the present invention;

FIG. 4 schematically illustrates a row merge process according to an embodiment of the present invention;

FIG. 5 is a block diagram illustrating a quasi-cyclic LDPC encoder according to an embodiment of the present invention;

FIG. 6 is a block diagram illustrating a quasi-cyclic LDPC code decoder included in a signal reception apparatus in an MMT system according to an embodiment of the present invention;

FIG. 7 is a flowchart illustrating an operation process of a quasi-cyclic LDPC code generator included in a signal transmission apparatus in an MMT system according to an embodiment of the present invention; and

FIG. 8 a flowchart illustrating an operation process of a quasi-cyclic LDPC code decoder included in a signal reception apparatus in an MMT system according to an embodiment of the present invention.

Throughout the drawings, the same drawing reference numerals will be understood to refer to the same elements, features and structures.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

Various embodiments of the present invention will now be described in detail with reference to the accompanying drawings. In the following description, specific details such as detailed configuration and components are merely provided to assist the overall understanding of these embodiments of the present invention. Therefore, it should be apparent to those skilled in the art that various changes and modifications of the embodiments described herein can be made without departing from the scope and spirit of the present invention. In addition, descriptions of well-known functions and constructions are omitted for clarity and conciseness.

Although embodiments of the present invention will be described below with reference to an MPEG MMT system, it will be understood by those of ordinary skill in the art that the present invention is also applicable to any one of a Long-Term Evolution (LTE) mobile communication system, a Long-Term Evolution Advanced (LTE-A) mobile communication system, and an Institute of Electrical and Electronics Engineers (IEEE) 802.16m communication system.

FIG. 1 is a block diagram illustrating a quasi-cyclic LDPC code generator included in a signal transmission apparatus in an MMT system according to an embodiment of the present invention.

Referring to FIG. 1, a quasi-cyclic LDPC code generator includes a quasi-cyclic LDPC encoder 111 and a parity check matrix generator 113. An information word vector is input to the quasi-cyclic LDPC encoder 111. The information word vector includes k information symbols. Additionally, control information including includes (k,n,m) information is input to the quasi-cyclic LDPC encoder 111, where k denotes a number of information word symbols included in the information word vector, n denotes a number of codeword symbols, i.e., quasi-cyclic LDPC codeword symbols, included in a codeword vector, i.e., a quasi-cyclic LDPC codeword vector, and m denotes a number of parity symbols included in a parity vector.

The parity check matrix generator 113 also receives the control information, converts a prestored base matrix into a parity check matrix, based on the control information, and outputs the converted parity check matrix to the quasi-cyclic LDPC encoder 111.

The quasi-cyclic LDPC encoder 111 generates a quasi-cyclic LDPC codeword vector by quasi-cyclic LDPC encoding the information word vector using the received control information and converted parity check matrix.

Although FIG. 1 illustrates the parity check matrix generator 113 generating the parity check matrix and outputting the parity check matrix to the quasi-cyclic LDPC encoder 111, alternatively, the quasi-cyclic LDPC encoder 111 may prestore the parity check matrix, and in this case, the parity check matrix generator 113 is not utilized.

Additionally, although FIG. 1 illustrates the control information being input from the outside to the quasi-cyclic LDPC encoder 111 and the parity check matrix generator 113, alternatively, the quasi-cyclic LDPC encoder 111 and the parity check matrix generator 113 may prestore the control information, and in this case, need not receive the control information from the outside.

Further, although FIG. 1 illustrates the quasi-cyclic LDPC encoder 111 and the parity check matrix generator 113 are shown in FIG. 1 as separate units, the quasi-cyclic LDPC encoder 111 and the parity check matrix generator 113 may also be incorporated into a single unit.

Although not illustrated, a signal transmission apparatus includes the quasi-cyclic LDPC code generator and a transmitter, and the quasi-cyclic LDPC code generator and the transmitter may be incorporated into a single unit.

FIG. 2 is a block diagram illustrating a parity check matrix generator according to an embodiment of the present invention.

Referring to FIG. 2, a parity check matrix generator 113 includes a parent parity check matrix generation unit 211, a parity check matrix generation unit 213, and a conversion information generation unit 215. The parent parity check matrix generation unit 211 reads a parent parity check matrix from an internal storage unit, or generates the parent parity check matrix using a preset scheme, and outputs the parent parity check matrix to the parity check matrix generation unit 213.

In accordance with an embodiment of the present invention, a parent parity check matrix Q is generated using a base matrix B, which includes K columns and M rows, and each of K columns maps to an information symbol block. Each element included in the base matrix B has a value of 0 or 1, and locations of elements with a value of 1 in each of the M rows may be expressed as a sequence, as shown in Equation (1).
R_i={j|0≦j<K,B_i,j=1}={r_i,0,r_i,1, . . . ,r_i,Di−1} (1)

In Equation (1), j denotes a column index, i denotes a row index, R_idenotes a sequence indicating locations on which elements have value 1 on the base matrix B, B_{i, j}denotes elements included in the base matrix B, and r_i,Di−1denotes elements included in the R_i. The D_idenotes a degree of the ith row.

For a base matrix B including 10 columns and 4 rows, the base matrix B may be expressed as shown in Equation (2).

$\begin{matrix} [\begin{matrix} 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \end{matrix}] & (2) \end{matrix}$

Each of R₀to R₃in Equation (2) is expressed as shown in Equation (3).
R₀={0,2,4,6}
R₁={1,3,5,7,9}
R₂={1,2,4,7,8}
R₃={0,5,6,9} (3)

The parent parity check matrix Q includes K×L columns and M×L rows, and is generated by substituting one of an L×L permutation matrix and an L×L zero matrix for each element B_i,jincluded in the base matrix B. The L×L permutation matrix is expressed as shown in Equation (4), and the L×L zero matrix is expressed as shown in Equation (5).

$\begin{matrix} Q_{i, j} = {[\begin{matrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \\ 1 & 0 & 0 & \dots & 0 \end{matrix}]}^{P} & (4) \end{matrix}$

In Equation (4), P denotes an exponent, a related permutation matrix is an identity matrix if P=0, and a related permutation matrix is a zero matrix if P=−1.

$\begin{matrix} Q_{i, j} = [\begin{matrix} 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 \end{matrix}] & (5) \end{matrix}$

For example, the parent parity check matrix Q may be expressed as shown in Equation (6).

$\begin{matrix} [\begin{matrix} 0 & - 1 & 0 & - 1 & 3 & - 1 & 3 & - 1 & - 1 & - 1 \\ - 1 & 2 & - 1 & 2 & - 1 & 1 & - 1 & 1 & - 1 & 0 \\ - 1 & 3 & 1 & - 1 & 0 & - 1 & - 1 & 2 & 3 & - 1 \\ 1 & - 1 & - 1 & - 1 & - 1 & 2 & 0 & - 1 & - 1 & 1 \end{matrix}] & (6) \end{matrix}$

In Equation (6), each permutation matrix is expressed as shown in Equation (7).
−1≦P_i,j<L (7)

In Equation (7), P_i,jdenotes an exponent of a permutation matrix arrayed on a location at which the ith row included in the base matrix B and the jth column included in the base matrix B crosses.

A location and an exponent of each of permutation matrixes included in the parent parity check matrix expressed in Equation (6) may be expressed as a sequence, as shown in Equation (8).

$\begin{matrix} \begin{matrix} T_{i} = {(j,) ❘ 0 \leq j < K, > - 1} \\ = {(t_{i, 0}, e_{i, 0}), (t_{i, 1}, e_{i, 1}), \dots, (Di - 1,)} \end{matrix} & (8) \end{matrix}$

In Equation (8), T_idenotes a sequence indicating locations at which permutation matrixes are arrayed on the parent parity check matrix, t_i,Di−1denotes a location of a permutation matrix included in the ith row among permutation matrixes included in the parent parity check matrix Q, and e_i,Di−1denotes an exponent of a permutation matrix arrayed on a related location, i.e., a location which t_i,Di−1indicates.

Each of T₀to T₃is expressed as shown in Equation (9).
T₀={(0,0),(2,0),(4,3),(6,3)}
T₁={(1,2),(3,2),(5,1),(7,1),(9,0)}
T₂={(1,3),(2,1),(4,0),(7,2),(8,3)}
T₃={(0,1),(5,2),(6,0),(9,1)} (9)

The parent parity check matrix generation unit 211 reads the base matrix B from an internal storage unit, or generates the base matrix B using a preset scheme.

As described above, the parent parity check matrix generated by the parent parity check matrix generation unit 211 only includes columns corresponding to an information symbol vector, i.e., an information part.

The conversion information generation unit 215 inputs control information, generates conversion information, and outputs the conversion information to the parity check matrix generation unit 213. The conversion information may include a scaling factor, a row separation factor, or a row merge factor, and a row separation pattern or a row merge pattern. If the conversion information includes the row separation factor, the row separation pattern is included in the conversion information.

If the conversion information includes the row merge factor, the row merge pattern is included in the conversion information. It will be understood by those of ordinary skill in the art that the row separation pattern and the merge pattern may be implemented in various formats.

A scaling factor S1 is used for determining a size of a parity check matrix to be generated by the parity check matrix generation unit 213. That is, the scaling factor S1 is used for changing a size of permutation matrixes and zero matrixes included in the parent parity check matrix Q. Here, S1 denotes a maximum integer satisfying k≦(K×L)/S1, where S1=2^a, and “a” denotes a maximum integer satisfying k≦(K×L)/S1.

If the size of the permutation matrixes and zero matrixes included in the parent parity check matrix Q is L×L, the size of the parity check matrix to be generated by the parity check matrix generation unit 213 is L′×L′ according to the scaling factor S1. Here, L′=L/S1.

A row separation factor S2 is used for separating rows included in the parent parity check matrix, when the parity check matrix generation unit 213 generates the parity check matrix. Here, S2=ceil(m/((M×L)/S1)), where S2=2^b, and b denotes a maximum integer satisfying 2^b≧ceil(m/((M×L/S1)).

For example, each of the rows included in the parent parity check matrix is separated into 2 rows, if the row separation factor S2 is 2.

FIG. 3 illustrates a row separation process according to an embodiment of the present invention.

Referring to FIG. 3, for the parent parity check matrix expressed in Equation (6), when a row separation factor S2 is 2, the first row 311 included in the parent parity check matrix is separated into 2 rows 313. In FIG. 3, C0 to C9 are indexes of each row of elements included in the first row 311, and C′0 to C′9 are indexes of each row of elements included in the 2 rows 313.

A matrix generated by performing a row separation operation and a scaling operation on a base matrix is equal to a matrix generated by performing the row separation operation on a parent parity check matrix, which is generated by performing the scaling operation on the base matrix. Therefore, a row separation pattern may be expressed using a row separated-base matrix or a row separated-parent parity check matrix.

A criterion for separating one row into n rows is shown in Equation (10).
1) T₀=T′₀∪T′₁∪ . . . ∪T′_n−1
2) ∩T′_j={ } for all 0≦≠j<n (10)

When Equation (10) is satisfied, one row is separated into n rows. That is, one row is separated into n rows expressed as shown in Equation (11).
T₀→T′₀,T′₁, . . . ,T′_n−1 (11)

The row merge factor S3 is used for merging rows included in the parent parity check matrix, when the parity check matrix generation unit 213 generates the parity check matrix. Here, S3=ceil((((M×L)/S1)/m)), S3=2^c, and c denotes a maximum integer satisfying 2^c≧ceil(((M×L)/S1)/m)).

For example, the rows included in the parent parity check matrix are merged on two-row basis, if the row merge factor S3 is 2.

FIG. 4 illustrates a row merge process according to an embodiment of the present invention.

Referring to FIG. 4, for the parent parity check matrix in Equation (6), when a row merge factor S3 is 3, the first row and the second row 411 included in the parent parity check matrix are merged into one row 413. In FIG. 4, C0 to C9 are indexes of each row of elements included in the first row and the second row 411, and C′0 to C′9 are indexes of each row of elements included in one row 413.

A matrix generated by performing a row merge operation and a scaling operation on a base matrix is equal to a matrix generated by performing the row merge operation on a parent parity check matrix, which is generated by performing the scaling operation on the base matrix. Therefore, a row merge pattern may be expressed using a row merged-base matrix or a row merged-parent parity check matrix.

A criterion for merging n rows into one row is expressed as shown in Equation (12).
T_i∩T_j={ } for all 0≦i≠j<n
T′₀=T₀∪T₁∪ . . . ∪T_n−1 (12)

When the criterion expressed in Equation (12) is satisfied, n rows are merged into one row. That is, n rows are merged into one row as shown in Equation (13).
T₀,T₁, . . . ,T_n−1→T′₀ (13)

If a row separation operation according to a row separation factor is performed, a row merge operation according to a row merge factor is not performed. If a row merge operation according to a row merge factor is performed, a row separation operation according to a row separation factor is not performed.

If a parity check matrix is generated according to a row merge factor or a row separation factor, a location and an exponent of each permutation matrix included in the parity check matrix are expressed as shown in Equation (14).
(t′_i,j,e′_i,j)=(t_i,j,f(e_i,j,L,S1)) (14)

The parity check matrix generation unit 213 converts the parent parity check matrix output from the parent parity check matrix generation unit 211 into a parity check matrix using the conversion information output from the conversion information generation unit 215.

If a parity check matrix generated by the parity check matrix generation unit 213 is H, the parity check matrix H includes H_Iand H_P, H_Iis a matrix corresponding to an information word vector, and H_Pis a matrix corresponding to a parity vector. H_Iis a parent parity check matrix generated by the parent parity check matrix generation unit 211, and a quasi-cyclic matrix. H_Pis an encoding matrix, and includes M′×L′ columns and M′×L′ rows. Here, H_Pneed not be a quasi-cyclic matrix, and may be one of a bit-wise dual diagonal (accumulator) matrix as shown in Equation (15), a block-wise dual diagonal matrix as shown in Equation (16), and a quasi-cyclic Block LDPC (BLDPC) used in an IEEE 802.16e communication system as shown in Equation (17).

If an information word vector includes k symbols and a parity vector includes m parity symbols, the parity check matrix generation unit 213 substitutes l×l (l=L/S1) permutation matrixes and l×l (l=L/S1) zero matrixes for permutation matrixes and zero matrixes included in a parent parity check matrix Q. The substituted permutation matrix is shown in Equation (18).
p_i,j=f(P_i,j,L,S1) (18)

The parity check matrix generation unit 213 generates H_Iincluding M block rows by separating each block row included in a conversion parent parity check matrix including the substituted permutation and zero matrixes, based on a row separation factor S2, or merging block rows included in the conversion parent parity check matrix matrixes, based on a row merge factor S3.

The parity check matrix generation unit 213 generates a parity check matrix as shown in Equation (19) by concatenating H_Iand H_P.
H=[H_I|H_p] (19)

In Equation (19), H_Pincludes M′×1 rows and M′×1 columns.

A permutation matrix is expressed in Equation (20).
1) f(P_i,j,L,S1)=floor(P_i,j/S1)
2) f(P_i,j,L,S1)=P_i,jmod(L/S1) (20)

Although FIG. 2 illustrates the parent parity check matrix generation unit 211 generating the parent parity check matrix, the conversion information generation unit 215 generating the conversion information, and the parity check matrix generation unit 213 generating the parity check matrix using the parent parity check matrix and the conversion information, alternatively, the parity check matrix generation unit 213 may prestore the parent parity check matrix and the conversion information, and in this case, the parent parity check matrix generation unit 211 and the conversion information generation unit 215 are not utilized.

Additionally, Although FIG. 2 illustrates the parent parity check matrix generation unit 211, the parity check matrix generation unit 213, and the conversion information generation unit 215 as separate units, these components may be incorporated into a single unit.

TABLE 1 information word symbol 10, 20, 40, 80 number (k) code rate (k/n) 10/12, 10/14, 10/16, 10/18

In Table 1, which shows design requirement criterion for an MMT system, the information word symbol number denotes a number of information word symbols that the MMT system should support. That is, the MMT system should support four types of information word symbol number, i.e., 10, 20, 40, and 80.

In Table 1, the code rate denotes the code rate that the MMT system should support. That is, the MMT system should support four types of code rate, i.e., 10/12, 10/14, 10/16, and 10/18.

The four types of code rate for each of the four types of information word symbol number should be supported, so 16 parity check matrixes will be used if the MMT system intends to design all of the quasi-cyclic LDPC codes satisfying the design requirement criterion described in Table 1.

The information word symbol number (k) 10, 20, 40, and 80, as described in Table 1, are in multiples one another, so a required parity symbol number for satisfying the code rate (k/n) 10/12, 10/14, 10/16, 10/18, as descried in Table 1, are also in multiples of one another if the information word symbol number is determined.

In accordance with an embodiment of the present invention, the design requirement criterion in Table 1 is satisfied by performing a scaling operation and a row separation operation on a parent parity check matrix including 16(=2×8) rows and 80(=10×8) columns, which is generated by substituting an 8×8 permutation matrix for each element included in a base matrix including 2 rows and 10 columns.

For a base matrix expressed as in Equation (21), when the number of rows is 2 and the number of the columns is 10, a parent parity check matrix, which is generated by substituting an 8×8 permutation matrix for each element included in the base matrix in Equation (21), is expressed as shown in Equation (22).
R₀={0,1,2,3,4}
R₁={5,6,7,8,9} (21)
T₀={(0,0),(1,7),(2,4),(3,1),(4,6)}
T₁={(5,3),(6,0),(7,1),(8,6),(9,7)} (22)

A method for supporting various information word symbol numbers using a scaling scheme will be described with reference to a case in which a criterion 2) in Equation (20) is applied to the parent parity check matrix expressed in Equation (22).

information word symbol numbers 10, 20, and 40 may be supported by applying a permutation matrix substitution, in which a scaling factor S1=8, 4, and 2 is applied to the parent parity check matrix expressed in Equation (22), respectively.

Matrixes generated by applying a permutation matrix substitution, expressed as criterion 2) in Equation (20), in which a scaling factor S1=2 and 4 is applied to the parent parity check matrix expressed in Equation (22), respectively, are expressed as shown in Equations (23) to (24).
T₀={(0,0),(1,3),(2,0),(3,1),(4,2)}
T₁={(5,3),(6,0),(7,1),(8,2),(9,3)} (23)
T₀={(0,0),(1,1),(2,0),(3,1),(4,0)}
T₁={(5,1),(6,0),(7,1),(8,0),(9,1)} (24)

Parent parity check matrixes that support information word symbol numbers 40 and 20 are expressed in Equations (23) to (24), respectively. If the number of information word symbols is 10, L′=L/8=1. Therefore, a parent parity check matrix is equal to a base matrix as expressed in Equation (21).

If the number of information word symbols is 80, parity symbol numbers that satisfy the design requirement criterion in Table 1 are 16, 32, 48, and 64 for 10/12, 10/14, 10/16, and 10/18, respectively. In an LDPC code, the number of parity symbols is equal to the number of parity check equations. Therefore, a code rate is 80/(80+16)= 10/12, if the parity check matrix in Equation (22) is used as a parent parity check matrix, and the code rates described in Table 1 are satisfied if each row included in the parity check matrix is separated into 2, 3, and 4 rows, respectively.

A row separation patter may be designed in various forms. However, in accordance with an embodiment of the present invention, a rule provided below is applied in order to provide a uniform degree of distribution for the columns in a parity check matrix, which is generated by applying a row separation scheme.

The jth row among rows that are generated by separating T_i={(t_i,0,e_i,0), (t_i,1,e_i,1), . . . , (Di−1,Di−1)} into n rows corresponds to the (nxi+j)th row, and x+j={()|0≦k<D_i, k mod n=(n−1−j)}, if all rows included in a parent parity check matrix are separated into n rows, respectively.

Matrixes generated by applying a row separation factor S2=2, 3, and 4 to the parent parity check matrix as shown in Equation (22), respectively, are expressed as shown in Equations (25) to (27).
T′₀={(1,1),(3,1)}
T′₁={(0,0),(2,0),(4,0)}
T′₂={(6,0),(8,0)}
T′₃={(5,1),(7,1),(9,1)} (25)

wherein Equation (25), T′₀, T′₁denote columns that are generated by separating T₀as shown in Equation (21), and T′₂and T′₃denote columns that are generated by separating T₁as shown in Equation (22).
T′₀={(2,4)}
T′₁={(1,7),(4,6)}
T′₂={(0,0),(3,1)}
T′₃={(7,1)}
T′₄={(6,0),(9,7)}
T′₅={(5,3),(8,6)} (26)

wherein Equation (26), T′₀, T′₁, and T′₂denote columns that are generated by separating T₀as shown in Equation (21), and T′₃, T′₄, and T′₅denote columns that are generated by separating T₁as shown in Equation (22).
T′₀={(3,1)}
T′₁={(2,4)}
T′₂={(1,7)}
T′₃={(0,0),(4,6)}
T′₄={(8,6)}
T′₅={(7,1)}
T′₆={(6,0)}
T′7={(5,3),(9,7)} (27)

wherein Equation (27), T′₀, T′₁, T′₂, and T′₃denote columns that are generated by separating T₀as shown in Equation (21), and T′₄, T′₅, T′₆, and T′₇denote columns that are generated by separating T₁as shown in Equation (22).

TABLE 2 information word symbol 400, 800, 1600, 3200, 6400 number (k) code rate (k/n) 20/21, 20/22, 20/23, 20/24

In Table 2, which shows design requirement criterion for an MMT system, the information word symbol number denotes a number of information word symbols that the MMT system should support. That is, the MMT system should support five types of information word symbol numbers, i.e., 400, 800, 1600, 3200, and 6400.

In Table 2, the code rate denotes the code rate that the MMT system should support. That is, the MMT system should support five types of code rates, i.e., 20/21, 20/22, 20/23, and 20/24.

A base matrix including 20 rows and 400 columns, which is expressed using Equation (1), is expressed in Equation (28).
R₀={1,2,3,5,6,7,9,12,14,15,19,20,26,34,35,38,45,46,48,56,57,62,63, 71,75,77,78,82,83,85,88,90,92,93,97,99,104,107,110,111,116,117,120,121, 125,127,128,129,131,134,150,152,156,158,159,161,163,164,165,168,171,172, 175,177,180,185,193,194,195,199,200,202,204,207,212,213,214,217,223,224, 226,227,228,232,236,240,241,245,250,251,255,260,267,268,272,273,275,276, 278,284,288,289,291,292,297,299,302,309,310,311,312,326,330,334,335,337, 338,340,342,343,347,349,350,351,357,361,364,365,367,369,373,375,376,377, 379,383,384,388,389,391}
R₁={2,5,8,10,12,13,14,17,23,24,29,30,33,37,45,46,47,56,60,65,73, 77,78,81,89,94,99,100,102,107,111,112,117,125,127,128,133,134,136,137, 138,141,143,155,157,158,160,161,163,169,170,174,176,177,178,180,182,186, 187,188,189,191,192,196,198,199,200,202,204,207,210,214,217,221,224,226, 228,233,236,239,241,246,249,251,256,257,259,263,264,266,267,270,271,280, 282,285,286,291,292,295,302,305,306,308,309,311,312,315,316,321,322,323, 324,327,328,338,342,343,346,347,349,356,361,363,367,369,372,373,374,376, 380,382,387,389,390,392,393,394,395,397}
R₂={1,2,3,5,6,12,13,15,19,20,21,22,33,36,38,39,40,43,44,46,47,48, 51,53,57,58,59,61,70,71,73,74,79,85,86,88,89,90,92,95,99,103,104,105,111, 115,128,130,133,136,139,142,145,155,156,160,168,171,176,182,183,185,186, 191,193,199,204,205,207,213,217,224,226,228,230,233,234,236,238,239,240, 246,247,248,249,250,251,253,254,258,262,264,268,270,271,274,277,280,286, 287,293,294,296,297,299,301,302,304,306,308,309,314,315,316,317,322,325, 326,327,331,333,334,335,345,348,351,358,360,362,363,371,375,376,378,379, 387,389,394,396,399}
R₃={2,3,4,7,9,14,15,18,22,29,30,32,36,40,50,53,54,55,60,64,68,70, 71,75,77,81,85,90,91,95,96,100,101,103,104,105,107,108,109,110,111,113, 116,121,123,124,131,132,133,136,137,140,144,145,149,152,155,159,162,164, 166,167,168,171,174,176,180,181,182,183,184,188,189,190,197,199,203,209, 213,215,223,232,234,235,240,244,247,248,253,254,255,256,257,260,265,266, 274,278,280,281,286,288,291,292,294,295,297,299,301,304,306,314,321,322, 325,327,334,336,338,346,348,351,352,353,355,364,366,369,370,371,373,374, 376,377,382,386,387,389,396,399}
R₄={0,1,5,16,17,18,19,21,26,28,29,31,34,36,37,39,40,44,50,55,56, 58,67,72,73,74,76,78,80,81,84,87,89,93,96,99,107,108,110,114,117,118,119, 120,123,125,128,132,138,140,143,147,152,155,164,167,171,173,174,175,180, 181,182,188,191,195,199,200,204,206,207,208,211,212,213,217,218,222,223, 228,230,231,241,242,249,253,254,255,256,257,259,261,262,264,265,268,269, 275,277,278,280,285,286,289,293,295,297,298,300,307,311,315,316,318,320, 324,326,338,339,342,346,347,348,356,357,358,359,360,361,363,364,365,368, 371,375,385,388,389,393,398}
R₅={1,6,8,12,18,19,21,24,25,31,34,35,37,38,40,41,43,44,45,49,53, 59,61,65,72,76,83,84,86,88,101,105,108,109,113,114,115,117,119,121,122, 124,129,131,132,135,136,137,139,141,142,144,148,151,154,159,166,169,175, 176,183,184,186,193,194,195,196,197,205,209,210,214,217,218,219,222,223, 225,227,228,229,232,235,236,241,243,245,247,248,250,262,263,266,267,272, 274,275,278,281,282,287,288,291,294,298,300,303,308,309,313,317,319,325, 327,329,332,333,335,336,338,350,351,353,354,355,356,359,360,362,363,366, 368,373,376,378,379,383,385,388,398}
R₆={4,6,7,11,13,15,16,22,27,28,31,33,34,36,38,42,43,44,54,55,57, 69,70,71,73,75,76,78,80,81,84,86,87,88,93,95,101,102,103,104,107,110,111, 112,114,117,119,120,122,130,131,134,135,138,139,147,149,150,153,155,165, 168,170,171,173,180,185,188,193,196,198,201,203,205,207,208,211,215,216, 217,220,227,229,231,233,234,237,241,242,248,258,261,262,263,264,266,268, 269,273,279,283,287,288,290,292,294,295,296,300,302,308,313,317,321,325, 326,332,336,341,342,343,355,356,358,365,369,370,371,377,385,387,388,390, 391,392,395,396,397,398,399}
R₇={0,1,3,4,10 15 17 18 20 22 25 27 31 33 36 38 41 45 48 52 55 59 62 63 64 66 69 74 77 80 82 84 87 99 103 107 108 110 118 121 122 125 129 131 132 137 138 139 142 145 146 147 148 151 153 154 157 165 168 173 175 179 183 184 185 186 187 189 194 196 203 204 205 206 209 215 219 222 224 225 226 229 231 232 234 237 238 242 243 245 247 248 250 251 253 255 256 259 262 263 267 268 269 270 271 276 281 285 286 289 290 292 293 297 306 308 309 314 317 318 320 322 324 328 336 339 340 341 346 347 348 360 367 368 372 375 378 382 390 396}
R₈={2,3,8,11,16,24,25,26,28,31,33,34,39,44,50,51,52,54,56,57,58, 60,63,64,66,67,68,69,70,72,80,83,90,95,97,98,99,100,101,103,105,108,109, 118,119,126,127,130,133,138,140,145,147,149,151,152,156,158,161,162,163, 166,169,170,177,178,179,181,186,190,193,196,201,202,203,205,207,209,211, 214,227,231,233,239,241,242,243,244,245,247,251,254,257,268,276,277,282, 284,285,288,292,295,304,305,307,308,310,312,314,317,318,323,324,328,329, 330,335,340,344,345,346,347,350,352,354,355,357,361,365,367,370,372,376, 377,380,382,386,389,392,394}
R₉={1,3,4,5,9,10,20,21,25,26,32,33,37,46,53,54,58,62,68,70,72,73, 74,80,81,82,84,86,89,91,92,96,97,99,102,111,115,116,118,123,132,138,139, 141,143,146,149,152,154,157,158,159,162,167,170,171,172,174,175,177,178, 181,184,190,195,196,198,199,200,201,202,204,207,208,216,220,226,229,230, 233,235,237,246,247,248,252,255,257,258,263,264,265,267,269,270,271,272, 275,279,286,293,296,298,302,307,311,313,316,319,320,330,331,332,333,335, 339,340,344,345,346,350,351,362,363,366,367,368,369,373,375,377,378,380, 384,387,390,392,394,395,397}
R₁₀={0,3,4,6,8,9,10,11,13,16,17,18,30,31,34,38,42,43,45,49,50,51, 52,53,58,60,61,62,67,69,74,79,87,88,91,92,93,97,98,101,104,108,111,112, 115,116,122,124,126,129,130,133,135,143,144,145,150,151,156,157,158,159, 161,165,170,171,173,174,182,184,191,192,198,204,206,209,216,219,220,222, 226,228,232,237,240,244,245,252,253,255,257,258,259,262,267,273,278,279, 290,292,296,299,301,303,310,315,319,320,323,324,326,330,331,332,333,336, 337,339,340,343,344,349,352,358,366,367,372,376,377,378,379,381,383,388, 389,391,393,394,397,399}
R₁₁={4,6,10,13,15,16,18,20,23,30,32,37,39,42,45,48,49,51,52,57,59, 61,66,73,75,78,82,83,87,90,92,93,95,96,97,98,106,108,109,113,115,116,119, 126,130,131,135,138,142,144,146,147,148,149,150,151,153,154,159,164,166, 167,168,169,170,178,180,181,183,186,192,193,197,198,202,203,210,212,213, 215,217,218,219,220,221,225,226,227,229,230,231,233,235,238,239,244,246, 249,250,256,262,265,275,276,279,280,283,286,291,296,300,303,309,311,312, 317,318,319,323,327,329,336,345,348,354,358,359,360,364,367,371,373,374, 375,388,390,391,394,397,398}
R₁₂={1,7,8,11,14,16,22,23,24,27,28,30,44,45,46,48,50,59,62,63,64, 65,66,70,76,80,82,83,84,85,86,94,97,98,100,101,102,109,112,113,114,117, 120,121,123,124,127,128,134,140,141,142,152,160,162,163,165,167,172,176, 177,178,179,180,181,185,187,189,190,192,198,202,206,208,218,219,221,224, 225,227,233,239,240,241,242,243,249,250,251,252,254,256,259,260,263,265, 270,272,273,277,278,284,287,290,293,301,307,311,312,313,316,321,322,323, 324,325,330,331,334,335,336,337,341,344,347,349,350,353,355,358,359,360, 369,380,383,384,391,392,393,397}
R₁₃={5,7,9,11,12,13,15,23,24,25,27,31,33,35,36,41,42,43,47,48,49, 51,52,53,58,65,69,71,81,85,86,88,94,96,97,102,104,106,113,114,115,120, 122,123,125,127,129,130,140,142,144,146,147,148,152,154,156,157,160,162, 165,166,173,175,176,179,183,190,192,193,195,199,205,206,208,209,210,212, 213,214,216,218,221,222,223,224,229,230,231,236,243,246,249,261,266,271, 273,276,281,282,283,284,289,290,293,298,300,304,305,307,312,314,315,316, 318,319,320,325,329,334,338,342,347,350,351,352,353,354,361,364,365,368, 371,372,380,382,384,385,386,399}
R₁₄={0,6,12,19,20,25,26,29,30,32,34,40,41,47,49,51,52,54,55,58,60, 63,64,66,69,72,75,77,78,79,81,82,87,90,91,92,94,95,100,105,106,109,110, 112,115,118,120,124,125,126,132,133,135,140,141,143,145,146,153,155,161, 163,167,172,175,178,183,184,186,187,194,197,201,206,208,211,216,219,223, 225,235,237,238,240,246,252,259,264,273,274,275,276,277,280,282,287,289, 293,294,298,300,301,310,313,315,316,318,319,320,321,322,327,328,330,331, 334,337,339,340,359,362,365,368,370,371,372,374,379,381,382,384,386,387, 388,391,393,395,396,398,399}
R₁₅={2,4,8,11,13,16,21,26,27,28,29,32,36,38,39,42,51,54,56,59,61, 63,65,67,68,71,72,74,79,82,84,87,88,94,102,106,112,116,118,121,124,126, 127,131,132,134,135,136,139,140,142,149,150,156,157,158,159,162,163,165, 166,172,173,174,179,181,184,185,189,191,194,195,197,201,203,208,209,212, 214,215,218,219,221,227,230,232,234,243,244,252,253,260,261,265,271,272, 273,281,282,283,285,288,289,291,294,303,306,310,317,318,319,323,329,330, 333,337,338,339,341,342,343,344,345,349,350,353,356,359,362,366,370,374, 377,379,381,383,384,390,391,395}
R₁₆={7,11,17,19,22,27,29,30,35,42,47,54,55,56,63,64,66,67,71,72, 73,75,76,77,79,80,85,89,90,93,94,96,98,100,106,113,116,118,122,126,128, 133,136,137,141,144,147,148,150,151,153,155,157,162,163,164,167,169,176, 182,185,187,189,190,191,197,203,210,211,212,216,218,220,224,225,228,232, 235,236,237,238,244,245,252,256,258,260,261,265,266,269,270,275,276,277, 279,280,284,285,288,290,295,297,298,299,300,301,302,303,304,305,308,311, 313,315,321,325,326,328,331,332,334,335,337,341,344,355,356,357,358,362, 366,373,374,381,385,386,395,397,398}
R₁₇={0,2,7,12,14,19,21,22,23,24,27,28,32,35,37,39,40,41,43,44,46, 47,53,57,62,64,65,68,77,79,83,85,86,91,92,94,95,103,104,105,106,107,109, 112,113,119,120,126,135,143,144,145,146,153,154,160,164,168,169,172,177, 178,179,188,190,192,194,195,196,200,201,213,215,229,230,231,235,242,243, 244,246,247,249,252,253,254,258,264,266,269,271,272,274,281,283,284,287, 289,298,299,304,305,306,307,309,320,321,322,327,328,329,331,333,343,345, 346,351,352,353,354,357,359,362,363,364,365,368,369,374,378,381,382,383, 384,385,386,390,393,396,399}
R₁₈={0,9,10,14,17,23,24,26,28,29,35,39,41,43,47,48,49,50,60,61,62, 65,66,67,68,69,70,74,75,76,79,89,91,93,98,100,103,106,114,122,123,124, 125,128,129,130,134,136,137,143,146,148,150,154,160,161,164,170,173,174, 177,182,187,188,191,192,200,202,210,211,215,216,220,221,222,223,225,234, 238,239,254,255,257,260,261,267,268,272,274,277,278,279,281,283,285,287, 290,294,296,297,303,304,305,307,310,313,314,323,326,328,329,332,337,341, 342,343,344,348,349,352,353,354,356,357,360,361,363,366,370,375,378,380, 381,385,386,387,392,393,396,398}
R₁₉={0,5,8,9,10,14,17,18,20,21,23,25,32,35,37,40,41,42,46,49,50, 52,55,56,57,59,60,61,67,68,76,78,83,89,91,96,98,101,102,105,110,114,117, 119,121,123,127,129,134,137,139,141,148,149,151,153,156,158,160,161,166, 169,172,179,187,188,189,194,197,198,200,201,205,206,210,211,212,214,220, 221,222,234,236,237,238,239,240,242,245,248,250,251,258,259,260,261,263, 269,270,274,279,282,283,284,291,295,296,299,301,302,303,305,306,310,312, 314,324,332,333,339,340,341,345,348,349,352,354,355,357,361,364,370,372, 379,380,381,383,392,394,395} (28)

In Table 2, when the maximum information word symbol number is 6400 and the number of columns included in the base matrix as shown in Equation (28) is 400, a required minimum parity symbol number is 320. Accordingly, a parent parity check matrix Q includes 400×16 columns and 20×16 rows.

A parent parity check matrix Q, which is expressed using Equation (8), is shown in Equation (29).
T₀={(1,8),(2,8),(3,10),(5,12),(6,8),(7,12),(9,8),(12,4),(14,12),(15,0),(19,0), (20,9),(26,4),(34,8),(35,1),(38,0),(45,13),(46,0),(48,13),(56,9),(57,3),(62,1),(63,8), (71,12),(75,8),(77,3),(78,2),(82,13),(83,13),(85,9),(88,1),(90,15),(92,4),(93,12), (97,0),(99,15),(104,5),(107,14),(110,13),(111,15),(116,9),(117,7),(120,9),(121,8), (125,15),(127,14),(128,15),(129,9),(131,5),(134,12),(150,12),(152,13),(156,1), (158,9),(159,13),(161,7),(163,5),(164,4),(165,13),(168,11),(171,9),(172,12), (175,12),(177,13),(180,5),(185,9),(193,1),(194,8),(195,9),(199,3),(200,9),(202,9), (204,3),(207,13),(212,13),(213,1),(214,13),(217,3),(223,13),(224,14),(226,10), (227,5),(228,7),(232,5),(236,14),(240,7),(241,7),(245,9),(250,12),(251,5),(255,5), (260,5),(267,4),(268,4),(272,4),(273,4),(275,9),(276,6),(278,5),(284,8),(288,1), (289,6),(291,2),(292,4),(297,12),(299,4),(302,4),(309,5),(310,4),(311,3),(312,5), (326,0),(330,10),(334,9),(335,4),(337,1),(338,6),(340,14),(342,10),(343,1),(347,4), (349,9),(350,1),(351,4),(357,14),(361,8),(364,0),(365,12),(367,6),(369,2),(373,4), (375,12),(376,12),(377,0),(379,0),(383,0),(384,1),(388,4),(389,0),(391,3)}
T₁={(2,4),(5,0),(8,9),(10,8),(12,8),(13,4),(14,8),(17,1),(23,12),(24,13), (29,9),(30,12),(33,9),(37,4),(45,0),(46,12),(47,8),(56,4),(60,0),(65,13),(73,13), (77,13),(78,12),(81,13),(89,12),(94,5),(99,5),(100,9),(102,9),(107,4),(111,5), (112,13),(117,4),(125,0),(127,12),(128,5),(133,12),(134,1),(136,1),(137,0),(138,12), (141,12),(143,4),(155,0),(157,12),(158,9),(160,0),(161,0),(163,8),(169,9),(170,12), (174,8),(176,4),(177,4),(178,8),(180,8),(182,8),(186,12),(187,8),(188,8),(189,8), (191,8),(192,1),(196,0),(198,8),(199,8),(200,0),(202,1),(204,4),(207,8),(210,1), (214,4),(217,2),(221,9),(224,0),(226,12),(228,9),(233,1),(236,8),(239,0),(241,0), (246,1),(249,1),(251,5),(256,8),(257,9),(259,9),(263,0),(264,1),(266,0),(267,1), (270,5),(271,9),(280,8),(282,0),(285,12),(286,1),(291,0),(292,8),(295,5),(302,12), (305,12),(306,9),(308,5),(309,4),(311,1),(312,13),(315,13),(316,1),(321,12),(322,5), (323,1),(324,5),(327,13),(328,5),(338,1),(342,12),(343,5),(346,13),(347,5),(349,0), (356,13),(361,1),(363,5),(367,5),(369,5),(372,5),(373,13),(374,4),(376,1),(380,4), (382,1),(387,5),(389,13),(390,5),(392,5),(393,1),(394,8),(395,9),(397,5)}
T₂={(1,2),(2,14),(3,3),(5,3),(6,10),(12,11),(13,3),(15,0),(19,13),(20,13), (21,6),(22,11),(33,10),(36,13),(38,6),(39,5),(40,12),(43,0),(44,3),(46,15),(47,3), (48,13),(51,7),(53,14),(57,15),(58,8),(59,6),(61,11),(70,7),(71,10),(73,10),(74,14), (79,1),(85,7),(86,14),(88,5),(89,5),(90,2),(92,10),(95,6),(99,14),(103,4),(104,6), (105,6),(111,10),(115,8),(128,3),(130,12),(133,4),(136,2),(139,6),(142,11),(145,14), (155,0),(156,14),(160,7),(168,5),(171,10),(176,12),(182,12),(183,8),(185,12), (186,9),(191,6),(193,15),(199,10),(204,0),(205,8),(207,5),(213,1),(217,4),(224,0), (226,0),(228,0),(230,4),(233,0),(234,12),(236,1),(238,8),(239,1),(240,3),(246,13), (247,12),(248,1),(249,8),(250,4),(251,1),(253,9),(254,0),(258,5),(262,3),(264,7), (268,6),(270,4),(271,1),(274,8),(277,5),(280,2),(286,11),(287,1),(293,3),(294,12), (296,7),(297,5),(299,7),(301,7),(302,11),(304,11),(306,3),(308,11),(309,15), (314,15),(315,8),(316,14),(317,10),(322,10),(325,11),(326,8),(327,15),(331,11), (333,1),(334,5),(335,9),(345,0),(348,13),(351,9),(358,9),(360,1),(362,15),(363,3), (371,3),(375,9),(376,7),(378,2),(379,1),(387,1),(389,11),(394,13),(396,15), (399,13)}
T₃={(2,4),(3,15),(4,7),(7,13),(9,8),(14,6),(15,5),(18,3),(22,4),(29,12),(30,7), (32,4),(36,14),(40,5),(50,5),(53,1),(54,11),(55,8),(60,8),(64,15),(68,12),(70,5), (71,9),(75,14),(77,2),(81,5),(85,10),(90,8),(91,11),(95,9),(96,11),(100,8),(101,5), (103,0),(104,4),(105,15),(107,13),(108,13),(109,13),(110,5),(111,9),(113,9),(116,0), (121,12),(123,12),(124,13),(131,9),(132,11),(133,1),(136,12),(137,11),(140,9), (144,12),(145,9),(149,0),(152,6),(155,7),(159,5),(162,0),(164,12),(166,6),(167,1), (168,9),(171,12),(174,5),(176,0),(180,5),(181,6),(182,0),(183,10),(184,11),(188,1), (189,1),(190,9),(197,6),(199,15),(203,6),(209,11),(213,15),(215,6),(223,3),(232,0), (234,6),(235,2),(240,10),(244,2),(247,6),(248,0),(253,6),(254,2),(255,2),(256,0), (257,10),(260,7),(265,2),(266,4),(274,0),(278,0),(280,4),(281,2),(286,2),(288,10), (291,2),(292,10),(294,1),(295,4),(297,0),(299,15),(301,2),(304,2),(306,3),(314,7), (321,1),(322,11),(325,10),(327,0),(334,11),(336,6),(338,8),(346,3),(348,3),(351,9), (352,8),(353,1),(355,11),(364,10),(366,10),(369,15),(370,14),(371,15),(373,13), (374,6),(376,15),(377,14),(382,5),(386,3),(387,2),(389,7),(396,8),(399,10)}
T₄={(0,0),(1,5),(5,9),(16,9),(17,1),(18,0),(19,0),(21,1),(26,0),(28,9),(29,8), (31,1),(34,7),(36,1),(37,5),(39,0),(40,15),(44,13),(50,9),(55,0),(56,0),(58,7), (67,15),(72,11),(73,1),(74,11),(76,5),(78,3),(80,3),(81,2),(84,0),(87,8),(89,5),(93,7), (96,2),(99,7),(107,14),(108,2),(110,0),(114,3),(117,1),(118,10),(119,10),(120,15), (123,11),(125,11),(128,10),(132,0),(138,10),(140,3),(143,2),(147,2),(152,14), (155,13),(164,13),(167,6),(171,2),(173,11),(174,8),(175,10),(180,7),(181,4), (182,14),(188,10),(191,2),(195,10),(199,9),(200,3),(204,14),(206,15),(207,0), (208,12),(211,6),(212,7),(213,7),(217,0),(218,15),(222,4),(223,2),(228,12),(230,9), (231,12),(241,6),(242,14),(249,15),(253,6),(254,1),(255,8),(256,1),(257,6),(259,11), (261,7),(262,12),(264,7),(265,15),(268,8),(269,12),(275,15),(277,11),(278,15), (280,11),(285,3),(286,13),(289,11),(293,13),(295,0),(297,3),(298,4),(300,12), (307,11),(311,1),(315,1),(316,6),(318,5),(320,4),(324,5),(326,9),(338,8),(339,13), (342,13),(346,5),(347,13),(348,7),(356,4),(357,2),(358,13),(359,2),(360,5),(361,0), (363,1),(364,8),(365,0),(368,13),(371,3),(375,8),(385,1),(388,3),(389,0),(393,5), (398,11)}
T₅={(1,14),(6,9),(8,11),(12,3),(18,10),(19,14),(21,9),(24,15),(25,10),(31,14), (34,11),(35,10),(37,8),(38,6),(40,7),(41,9),(43,3),(44,4),(45,14),(49,7),(53,7),(59,5), (61,13),(65,0),(72,11),(76,10),(83,15),(84,5),(86,11),(88,12),(101,5),(105,14), (108,5),(109,6),(113,6),(114,14),(115,13),(117,3),(119,0),(121,8),(122,7),(124,4), (129,1),(131,14),(132,1),(135,9),(136,8),(137,10),(139,3),(141,8),(142,15),(144,0), (148,12),(151,7),(154,11),(159,12),(166,1),(169,15),(175,3),(176,0),(183,13), (184,1),(186,3),(193,13),(194,3),(195,4),(196,3),(197,4),(205,4),(209,15),(210,0), (214,5),(217,10),(218,14),(219,4),(222,3),(223,0),(225,5),(227,9),(228,9),(229,9), (232,5),(235,5),(236,3),(241,1),(243,8),(245,0),(247,4),(248,5),(250,0),(262,8), (263,8),(266,5),(267,1),(272,1),(274,10),(275,1),(278,15),(281,0),(282,8),(287,15), (288,14),(291,14),(294,10),(298,1),(300,11),(303,10),(308,2),(309,10),(313,8), (317,0),(319,0),(325,2),(327,1),(329,2),(332,4),(333,2),(335,2),(336,0),(338,11), (350,10),(351,4),(353,3),(354,5),(355,5),(356,10),(359,6),(360,1),(362,15),(363,2), (366,2),(368,14),(373,6),(376,10),(378,12),(379,6),(383,1),(385,7),(388,14), (398,15)}
T₆={(4,8),(6,3),(7,10),(11,15),(13,11),(15,3),(16,11),(22,5),(27,3),(28,12), (31,12),(33,1),(34,13),(36,13),(38,13),(42,8),(43,12),(44,1),(54,14),(55,11),(57,12), (69,9),(70,1),(71,9),(73,14),(75,9),(76,8),(78,10),(80,8),(81,1),(84,9),(86,10), (87,13),(88,13),(93,0),(95,11),(101,0),(102,1),(103,1),(104,9),(107,1),(110,4), (111,1),(112,9),(114,9),(117,2),(119,1),(120,1),(122,8),(130,1),(131,1),(134,4), (135,10),(138,9),(139,15),(147,0),(149,3),(150,11),(153,2),(155,10),(165,13), (168,7),(170,11),(171,3),(173,8),(180,11),(185,3),(188,0),(193,10),(196,3),(198,1), (201,10),(203,11),(205,7),(207,14),(208,13),(211,4),(215,2),(216,7),(217,8),(220,3), (227,14),(229,5),(231,5),(233,14),(234,0),(237,6),(241,6),(242,6),(248,0),(258,6), (261,3),(262,14),(263,5),(264,2),(266,1),(268,12),(269,6),(273,14),(279,2),(283,14), (287,15),(288,0),(290,6),(292,2),(294,1),(295,10),(296,4),(300,4),(302,11),(308,12), (313,10),(317,6),(321,14),(325,1),(326,0),(332,10),(336,4),(341,10),(342,7),(343,7), (355,13),(356,1),(358,11),(365,7),(369,9),(370,3),(371,3),(377,2),(385,2),(387,4), (388,2),(390,7),(391,8),(392,5),(395,2),(396,1),(397,5),(398,7),(399,3)}
T₇={(0,15),(1,1),(3,6),(4,8),(10,9),(15,15),(17,10),(18,14),(20,9),(22,8), (25,14),(27,14),(31,4),(33,10),(36,3),(38,14),(41,10),(45,11),(48,0),(52,14),(55,10), (59,3),(62,12),(63,14),(64,3),(66,14),(69,0),(74,6),(77,5),(80,2),(82,4),(84,0), (87,6),(99,8),(103,2),(107,13),(108,10),(110,1),(118,12),(121,2),(122,7),(125,3), (129,6),(131,7),(132,7),(137,10),(138,6),(139,1),(142,15),(145,7),(146,3),(147,5), (148,11),(151,2),(153,9),(154,10),(157,0),(165,3),(168,0),(173,11),(175,15),(179,9), (183,8),(184,3),(185,1),(186,2),(187,2),(189,6),(194,9),(196,4),(203,11),(204,3), (205,11),(206,5),(209,1),(215,6),(219,1),(222,1),(224,0),(225,1),(226,1),(229,3), (231,7),(232,13),(234,0),(237,2),(238,2),(242,10),(243,11),(245,5),(247,1),(248,9), (250,2),(251,3),(253,0),(255,9),(256,4),(259,1),(262,9),(263,0),(267,3),(268,1), (269,0),(270,9),(271,8),(276,9),(281,9),(285,9),(286,10),(289,11),(290,8),(292,5), (293,10),(297,1),(306,0),(308,1),(309,1),(314,0),(317,3),(318,12),(320,13),(322,12), (324,12),(328,9),(336,12),(339,13),(340,7),(341,0),(346,15),(347,15),(348,11), (360,10),(367,0),(368,5),(372,11),(375,3),(378,6),(382,4),(390,6),(396,11)}
T₈={(2,8),(3,13),(8,11),(11,9),(16,14),(24,3),(25,1),(26,4),(28,12),(31,4), (33,5),(34,2),(39,4),(44,0),(50,3),(51,8),(52,14),(54,8),(56,7),(57,2),(58,8),(60,1), (63,12),(64,2),(66,3),(67,1),(68,12),(69,5),(70,7),(72,7),(80,9),(83,12),(90,9), (95,13),(97,3),(98,4),(99,12),(100,8),(101,2),(103,8),(105,2),(108,8),(109,5), (118,4),(119,5),(126,11),(127,5),(130,2),(133,0),(138,9),(140,4),(145,0),(147,1), (149,6),(151,6),(152,14),(156,1),(158,14),(161,4),(162,2),(163,1),(166,1),(169,1), (170,10),(177,2),(178,8),(179,0),(181,2),(186,4),(190,0),(193,3),(196,14),(201,2), (202,1),(203,0),(205,0),(207,11),(209,2),(211,2),(214,3),(227,3),(231,9),(233,9), (239,10),(241,2),(242,6),(243,8),(244,14),(245,13),(247,9),(251,0),(254,8),(257,0), (268,7),(276,10),(277,8),(282,10),(284,10),(285,3),(288,7),(292,4),(295,0),(304,12), (305,14),(307,13),(308,10),(310,7),(312,6),(314,12),(317,14),(318,13),(323,11), (324,11),(328,6),(329,10),(330,11),(335,11),(340,13),(344,14),(345,5),(346,7), (347,6),(350,13),(352,7),(354,3),(355,4),(357,3),(361,14),(365,5),(367,3),(370,12), (372,9),(376,4),(377,7),(380,12),(382,6),(386,2),(389,9),(392,0),(394,7)}
T₉={(1,2),(3,11),(4,1),(5,15),(9,7),(10,0),(20,2),(21,2),(25,8),(26,1),(32,6), (33,10),(37,4),(46,2),(53,7),(54,8),(58,3),(62,5),(68,0),(70,0),(72,0),(73,9),(74,3), (80,10),(81,2),(82,10),(84,13),(86,6),(89,0),(91,4),(92,5),(96,10),(97,5),(99,2), (102,13),(111,1),(115,5),(116,5),(118,4),(123,10),(132,6),(138,8),(139,2),(141,7), (143,7),(146,14),(149,2),(152,4),(154,4),(157,3),(158,1),(159,13),(162,0),(167,0), (170,6),(171,4),(172,4),(174,1),(175,1),(177,9),(178,14),(181,4),(184,12),(190,9), (195,8),(196,0),(198,12),(199,0),(200,0),(201,0),(202,12),(204,14),(207,3),(208,5), (216,15),(220,8),(226,0),(229,15),(230,12),(233,6),(235,8),(237,8),(246,15),(247,1), (248,7),(252,10),(255,12),(257,7),(258,3),(263,2),(264,14),(265,11),(267,2),(269,0), (270,8),(271,11),(272,8),(275,10),(279,6),(286,2),(293,0),(296,10),(298,14),(302,2), (307,7),(311,3),(313,15),(316,8),(319,14),(320,2),(330,10),(331,13),(332,8),(333,2), (335,6),(339,9),(340,2),(344,6),(345,10),(346,6),(350,10),(351,10),(362,14),(363,4), (366,14),(367,4),(368,1),(369,2),(373,0),(375,1),(377,3),(378,11),(380,0),(384,11), (387,4),(390,10),(392,3),(394,1),(395,0),(397,11)}
T₁₀={(0,10),(3,2),(4,6),(6,11),(8,0),(9,10),(10,0),(11,4),(13,12),(16,14), (17,0),(18,2),(30,10),(31,6),(34,8),(38,2),(42,7),(43,0),(45,2),(49,0),(50,8),(51,12), (52,6),(53,6),(58,10),(60,2),(61,8),(62,6),(67,0),(69,8),(74,2),(79,2),(87,4),(88,0), (91,6),(92,8),(93,2),(97,0),(98,10),(101,14),(104,2),(108,10),(111,10),(112,2), (115,2),(116,10),(122,2),(124,14),(126,8),(129,8),(130,0),(133,10),(135,2),(143,0), (144,8),(145,8),(150,8),(151,10),(156,10),(157,0),(158,10),(159,10),(161,0),(165,8), (170,2),(171,8),(173,4),(174,2),(182,14),(184,2),(191,14),(192,14),(198,4),(204,0), (206,15),(209,2),(216,12),(219,6),(220,6),(222,2),(226,6),(228,2),(232,8),(237,2), (240,14),(244,12),(245,0),(252,14),(253,10),(255,8),(257,10),(258,14),(259,4), (262,8),(267,2),(273,14),(278,6),(279,8),(290,10),(292,2),(296,6),(299,10),(301,2), (303,14),(310,6),(315,0),(319,6),(320,6),(323,8),(324,2),(326,6),(330,4),(331,4), (332,0),(333,8),(336,8),(337,10),(339,0),(340,0),(343,4),(344,4),(349,0),(352,8), (358,8),(366,8),(367,14),(372,0),(376,4),(377,14),(378,12),(379,4),(381,10), (383,12),(388,0),(389,8),(391,8),(393,6),(394,0),(397,12),(399,4),(400,0)}
T₁₁={(4,2),(6,8),(10,10),(13,10),(15,10),(16,2),(18,10),(20,10),(23,6),(30,0), (32,10),(37,2),(39,14),(42,14),(45,0),(48,12),(49,2),(51,7),(52,6),(57,11),(59,11), (61,1),(66,0),(73,3),(75,14),(78,11),(82,3),(83,2),(87,13),(90,10),(92,11),(93,11), (95,1),(96,15),(97,3),(98,5),(106,7),(108,0),(109,3),(113,4),(115,7),(116,1),(119,1), (126,7),(130,14),(131,3),(135,5),(138,1),(142,7),(144,5),(146,9),(147,11),(148,0), (149,1),(150,3),(151,3),(153,3),(154,3),(159,1),(164,9),(166,11),(167,3),(168,3), (169,7),(170,9),(178,3),(180,1),(181,5),(183,1),(186,5),(192,3),(193,3),(197,3), (198,5),(202,3),(203,13),(210,9),(212,9),(213,3),(215,1),(217,1),(218,15),(219,5), (220,3),(221,1),(225,1),(226,3),(227,6),(229,12),(230,8),(231,2),(233,2),(235,10), (238,2),(239,1),(244,6),(246,5),(249,11),(250,9),(256,1),(262,7),(265,2),(275,2), (276,8),(279,14),(280,0),(283,8),(286,1),(291,0),(296,2),(300,2),(303,2),(309,0), (311,0),(312,0),(317,0),(318,12),(319,0),(323,0),(327,0),(329,0),(336,0),(345,8), (348,8),(354,8),(358,8),(359,12),(360,8),(364,0),(367,10),(371,4),(373,0),(374,0), (375,8),(388,1),(390,10),(391,10),(394,2),(397,0),(398,10)}
T₁₂={(1,3),(7,3),(8,1),(11,13),(14,2),(16,1),(22,11),(23,0),(24,15),(27,8), (28,9),(30,11),(44,9),(45,7),(46,3),(48,7),(50,3),(59,1),(62,3),(63,3),(64,3),(65,7), (66,7),(70,2),(76,1),(80,3),(82,5),(83,7),(84,9),(85,11),(86,5),(94,11),(97,5),(98,9), (100,15),(101,1),(102,3),(109,0),(112,9),(113,9),(114,7),(117,5),(120,1),(121,1), (123,1),(124,3),(127,11),(128,1),(134,3),(140,9),(141,9),(142,5),(152,1),(160,5), (162,3),(163,9),(165,1),(167,11),(172,5),(176,1),(177,0),(178,1),(179,6),(180,2), (181,1),(185,0),(187,7),(189,0),(190,2),(192,15),(198,6),(202,8),(206,2),(208,10), (218,0),(219,2),(221,4),(224,3),(225,0),(227,6),(233,6),(239,3),(240,2),(241,0), (242,0),(243,6),(249,4),(250,4),(251,8),(252,8),(254,8),(256,0),(259,1),(260,1), (263,4),(265,0),(270,1),(272,0),(273,4),(277,2),(278,2),(284,4),(287,2),(290,2), (293,6),(301,10),(307,10),(311,2),(312,6),(313,12),(316,10),(321,10),(322,12), (323,12),(324,0),(325,0),(330,8),(331,2),(334,10),(335,10),(336,8),(337,10), (341,14),(344,10),(347,10),(349,6),(350,4),(353,8),(355,0),(358,2),(359,12),(360,1), (369,2),(380,4),(383,14),(384,11),(391,3),(392,2),(393,13),(397,3)}
T₁₃={(5,3),(7,0),(9,1),(11,11),(12,3),(13,1),(15,1),(23,3),(24,3),(25,3), (27,1),(31,9),(33,9),(35,9),(36,9),(41,11),(42,3),(43,3),(47,1),(48,7),(49,11),(51,13), (52,1),(53,3),(58,5),(65,15),(69,13),(71,1),(81,3),(85,9),(86,9),(88,7),(94,11),(96,3), (97,2),(102,7),(104,15),(106,7),(113,10),(114,2),(115,3),(120,2),(122,1),(123,3), (125,5),(127,1),(129,1),(130,4),(140,3),(142,11),(144,1),(146,3),(147,5),(148,15), (152,3),(154,0),(156,2),(157,9),(160,10),(162,6),(165,12),(166,14),(173,8),(175,8), (176,2),(179,11),(183,6),(190,9),(192,10),(193,4),(195,7),(199,0),(205,2),(206,1), (208,6),(209,0),(210,10),(212,2),(213,8),(214,0),(216,10),(218,1),(221,10),(222,0), (223,8),(224,10),(229,10),(230,0),(231,2),(236,2),(243,9),(246,0),(249,13),(261,10), (266,4),(271,0),(273,6),(276,10),(281,6),(282,0),(283,0),(284,8),(289,12),(290,0), (293,2),(298,10),(300,0),(304,8),(305,8),(307,4),(312,0),(314,4),(315,0),(316,2), (318,10),(319,12),(320,8),(325,12),(329,0),(334,1),(338,3),(342,4),(347,9),(350,1), (351,2),(352,2),(353,0),(354,11),(361,9),(364,13),(365,11),(368,9),(371,3),(372,0), (380,3),(382,4),(384,0),(385,1),(386,1),(399,4)}
T₁₄={(0,8),(6,0),(12,0),(19,2),(20,0),(25,8),(26,10),(29,0),(30,2),(32,0), (34,0),(40,8),(41,0),(47,0),(49,0),(51,2),(52,2),(54,8),(55,0),(58,2),(60,0),(63,0), (64,0),(66,3),(69,1),(72,8),(75,9),(77,0),(78,0),(79,5),(81,9),(82,12),(87,4),(90,1), (91,15),(92,11),(94,1),(95,0),(100,11),(105,10),(106,10),(109,11),(110,2),(112,15), (115,1),(118,9),(120,3),(124,13),(125,0),(126,15),(132,3),(133,11),(135,3),(140,11), (141,2),(143,9),(145,9),(146,11),(153,7),(155,1),(161,11),(163,7),(167,8),(172,1), (175,15),(178,1),(183,0),(184,8),(186,1),(187,11),(194,3),(197,11),(201,3),(206,3), (208,13),(211,3),(216,3),(219,1),(223,3),(225,11),(235,9),(237,9),(238,3),(240,3), (246,11),(252,9),(259,1),(264,3),(273,2),(274,9),(275,9),(276,5),(277,11),(280,9), (282,9),(287,1),(289,5),(293,9),(294,0),(298,2),(300,3),(301,1),(310,1),(313,0), (315,10),(316,12),(318,0),(319,12),(320,8),(321,10),(322,1),(327,1),(328,2),(330,0), (331,0),(334,2),(337,4),(339,0),(340,2),(359,1),(362,8),(365,5),(368,8),(370,10), (371,1),(372,2),(374,2),(379,1),(381,8),(382,8),(384,4),(386,4),(387,0),(388,4), (391,3),(393,0),(395,2),(396,2),(398,10),(399,0)}
T₁₅={(2,2),(4,10),(8,8),(11,0),(13,0),(16,0),(21,8),(26,2),(27,10),(28,2), (29,0),(32,3),(36,0),(38,2),(39,11),(42,1),(51,2),(54,10),(56,0),(59,0),(61,0),(63,1), (65,3),(67,11),(68,0),(71,3),(72,3),(74,10),(79,1),(82,0),(84,8),(87,1),(88,3),(94,8), (102,1),(106,9),(112,11),(116,8),(118,0),(121,11),(124,3),(126,9),(127,1),(131,9), (132,0),(134,1),(135,1),(136,1),(139,1),(140,1),(142,1),(149,1),(150,1),(156,1), (157,1),(158,3),(159,1),(162,11),(163,1),(165,9),(166,2),(172,0),(173,1),(174,3), (179,1),(181,1),(184,1),(185,1),(189,1),(191,3),(194,9),(195,9),(197,0),(201,1), (203,3),(208,1),(209,8),(212,8),(214,3),(215,11),(218,11),(219,8),(221,1),(227,11), (230,3),(232,3),(234,3),(243,9),(244,11),(252,2),(253,1),(260,0),(261,11),(265,9), (271,11),(272,1),(273,1),(281,10),(282,2),(283,9),(285,8),(288,10),(289,8),(291,11), (294,2),(303,3),(306,10),(310,2),(317,2),(318,11),(319,9),(323,3),(329,1),(330,1), (333,2),(337,8),(338,2),(339,2),(341,10),(342,0),(343,2),(344,0),(345,0),(349,10), (350,0),(353,8),(356,0),(359,0),(362,3),(366,8),(370,0),(374,10),(377,0),(379,10), (381,8),(383,8),(384,0),(390,2),(391,8),(395,0)}
T₁₆={(7,4),(11,5),(17,3),(19,6),(22,5),(27,6),(29,6),(30,7),(35,14),(42,6), (47,14),(54,4),(55,4),(56,2),(63,1),(64,6),(66,0),(67,2),(71,13),(72,0),(73,0), (75,12),(76,10),(77,4),(79,0),(80,0),(85,7),(89,6),(90,4),(93,6),(94,3),(96,5),(98,4), (100,0),(106,2),(113,14),(116,4),(118,2),(122,2),(126,12),(128,6),(133,6),(136,10), (137,14),(141,2),(144,14),(147,4),(148,2),(150,0),(151,0),(153,14),(155,2),(157,2), (162,2),(163,7),(164,2),(167,1),(169,3),(176,1),(182,12),(185,4),(187,3),(189,0), (190,1),(191,11),(197,1),(203,5),(210,12),(211,0),(212,0),(216,0),(218,3),(220,11), (224,4),(225,7),(228,10),(232,3),(235,8),(236,0),(237,1),(238,9),(244,7),(245,1), (252,0),(256,1),(258,3),(260,7),(261,1),(265,7),(266,1),(269,5),(270,1),(275,3), (276,2),(277,5),(279,0),(280,7),(284,7),(285,9),(288,1),(290,1),(295,7),(297,9), (298,1),(299,3),(300,3),(301,0),(302,1),(303,1),(304,3),(305,1),(308,1),(311,0), (313,1),(315,1),(321,5),(325,2),(326,5),(328,5),(331,10),(332,13),(334,1),(335,7), (337,5),(341,5),(344,4),(355,4),(356,5),(357,5),(358,4),(362,6),(366,5),(373,4), (374,4),(381,1),(385,1),(386,13),(395,4),(397,5),(398,6)}
T₁₇={(0,1),(2,5),(7,2),(12,1),(14,5),(19,1),(21,5),(22,5),(23,7),(24,3),(27,2), (28,9),(32,1),(35,13),(37,1),(39,1),(40,1),(41,0),(43,1),(44,10),(46,1),(47,0),(53,5), (57,1),(62,11),(64,5),(65,5),(68,5),(77,7),(79,1),(83,2),(85,1),(86,4),(91,1),(92,3), (94,4),(95,4),(103,3),(104,0),(105,4),(106,8),(107,5),(109,7),(112,7),(113,2), (119,5),(120,7),(126,1),(135,1),(143,2),(144,4),(145,12),(146,13),(153,5),(154,5), (160,4),(164,6),(168,0),(169,2),(172,0),(177,13),(178,4),(179,7),(188,0),(190,4), (192,8),(194,8),(195,4),(196,0),(200,0),(201,5),(213,11),(215,4),(229,1),(230,4), (231,0),(235,0),(242,4),(243,0),(244,4),(246,6),(247,4),(249,4),(252,4),(253,4), (254,4),(258,0),(264,0),(266,4),(269,7),(271,0),(272,0),(274,4),(281,2),(283,0), (284,0),(287,6),(289,2),(298,7),(299,10),(304,1),(305,3),(306,4),(307,7),(309,6), (320,14),(321,1),(322,7),(327,7),(328,6),(329,2),(331,10),(333,6),(343,0),(345,3), (346,0),(351,1),(352,11),(353,5),(354,2),(357,2),(359,1),(362,0),(363,3),(364,7), (365,6),(368,5),(369,5),(374,0),(378,5),(381,2),(382,7),(383,3),(384,12),(385,6), (386,0),(390,3),(393,3),(396,1),(399,1)}
T₁₈={(0,2),(9,3),(10,6),(14,2),(17,6),(23,7),(24,7),(26,2),(28,5),(29,12), (35,5),(39,4),(41,5),(43,3),(47,7),(48,3),(49,4),(50,0),(60,2),(61,2),(62,4),(65,0), (66,5),(67,14),(68,2),(69,4),(70,4),(74,4),(75,6),(76,2),(79,2),(89,6),(91,4),(93,6), (98,7),(100,4),(103,4),(106,1),(114,0),(122,4),(123,6),(124,14),(125,14),(128,0), (129,4),(130,4),(134,6),(136,4),(137,4),(143,5),(146,5),(148,4),(150,5),(154,0), (160,4),(161,4),(164,0),(170,4),(173,0),(174,1),(177,4),(182,5),(187,1),(188,6), (191,0),(192,1),(200,7),(202,0),(210,3),(211,9),(215,1),(216,1),(220,0),(221,1), (222,5),(223,3),(225,0),(234,5),(238,0),(239,5),(254,0),(255,5),(257,1),(260,13), (261,7),(267,7),(268,1),(272,5),(274,3),(277,3),(278,5),(279,1),(281,1),(283,6), (285,3),(287,0),(290,1),(294,1),(296,1),(297,1),(303,1),(304,0),(305,5),(307,5), (310,0),(313,2),(314,1),(323,3),(326,1),(328,2),(329,0),(332,1),(337,0),(341,0), (342,3),(343,3),(344,1),(348,1),(349,1),(352,1),(353,2),(354,1),(356,9),(357,0), (360,1),(361,13),(363,1),(366,0),(370,5),(375,0),(378,5),(380,1),(381,4),(385,0), (386,5),(387,4),(392,6),(393,7),(396,5),(398,5)}
T₁₉={(0,4),(5,12),(8,4),(9,8),(10,12),(14,8),(17,0),(18,10),(20,4),(21,9), (23,13),(25,5),(32,9),(35,12),(37,0),(40,12),(41,8),(42,0),(46,1),(49,1),(50,13), (52,12),(55,9),(56,0),(57,6),(59,14),(60,4),(61,10),(67,0),(68,0),(76,4),(78,6), (83,0),(89,6),(91,8),(96,0),(98,0),(101,8),(102,6),(105,1),(110,2),(114,5),(117,15), (119,1),(121,10),(123,5),(127,15),(129,0),(134,7),(137,9),(139,10),(141,1),(148,13), (149,4),(151,9),(153,4),(156,10),(158,11),(160,1),(161,11),(166,1),(169,12),(172,2), (179,11),(187,6),(188,9),(189,2),(194,5),(197,0),(198,1),(200,7),(201,13),(205,2), (206,13),(210,9),(211,1),(212,3),(214,11),(220,1),(221,2),(222,1),(234,1),(236,7), (237,8),(238,1),(239,3),(240,1),(242,7),(245,13),(248,13),(250,5),(251,7),(258,5), (259,3),(260,7),(261,14),(263,13),(269,5),(270,9),(274,0),(279,5),(282,6),(283,4), (284,0),(291,5),(295,3),(296,11),(299,5),(301,6),(302,2),(303,5),(305,4),(306,5), (310,9),(312,4),(314,13),(324,5),(332,12),(333,6),(339,7),(340,13),(341,14), (345,13),(348,12),(349,0),(352,12),(354,0),(355,1),(357,8),(361,0),(364,13), (370,13),(372,14),(379,12),(380,4),(381,11),(383,5),(392,10),(394,10),(395,8)} (29)

As described in Table 1, the MMT system may support the number of information word symbols 3200, 1600, 800, and 400 by performing a scaling factor S1=2, 4, 8, and 16 on the parent parity check matrix as shown in Equation (29).

If each row included in the parent parity check matrix as shown in Equation (29) is separated into 2, 3, and 4 rows, respectively, the design requirement criterion described in Table 2 may be satisfied.

Matrixes that are generated by applying the row separation rule (n=2, 3, and 4) described with reference to Table 1 to the parent parity check matrix as shown in Equation (29) are as shown in Equation (30) to Equation (32).

Therefore, code rates that the matrixes expressed as Equation (30) to Equation (32), i.e., the separated parent parity check matrixes, support are 20/22, 20/23, and 20/24, respectively.
T′₀={(2,8),(5,12),(7,12),(12,4),(15,0),(20,9),(34,8),(38,0),(46,0),(56,9), (62,1),(71,12),(77,3),(82,13),(85,9),(90,15),(93,12),(99,15),(107,14),(111,15), (117,7),(121,8),(127,14),(129,9),(134,12),(152,13),(158,9),(161,7),(164,4),(168,11), (172,12),(177,13),(185,9),(194,8),(199,3),(202,9),(207,13),(213,1),(217,3),(224,14), (227,5),(232,5),(240,7),(245,9),(251,5),(260,5),(268,4),(273,4),(276,6),(284,8), (289,6),(292,4),(299,4),(309,5),(311,3),(326,0),(334,9),(337,1),(340,14),(343,1), (349,9),(351,4),(361,8),(365,12),(369,2),(375,12),(377,0),(383,0),(388,4),(391,3)}
T′₁={(1,8),(3,10),(6,8),(9,8),(14,12),(19,0),(26,4),(35,1),(45,13),(48,13), (57,3),(63,8),(75,8),(78,2),(83,13),(88,1),(92,4),(97,0),(104,5),(110,13),(116,9), (120,9),(125,15),(128,15),(131,5),(150,12),(156,1),(159,13),(163,5),(165,13), (171,9),(175,12),(180,5),(193,1),(195,9),(200,9),(204,3),(212,13),(214,13),(223,13), (226,10),(228,7),(236,14),(241,7),(250,12),(255,5),(267,4),(272,4),(275,9),(278,5), (288,1),(291,2),(297,12),(302,4),(310,4),(312,5),(330,10),(335,4),(338,6),(342,10), (347,4),(350,1),(357,14),(364,0),(367,6),(373,4),(376,12),(379,0),(384,1),(389,0)}
T′₂={(5,0),(10,8),(13,4),(17,1),(24,13),(30,12),(37,4),(46,12),(56,4),(65,13), (77,13),(81,13),(94,5),(100,9),(107,4),(112,13),(125,0),(128,5),(134,1),(137,0), (141,12),(155,0),(158,9),(161,0),(169,9),(174,8),(177,4),(180,8),(186,12),(188,8), (191,8),(196,0),(199,8),(202,1),(207,8),(214,4),(221,9),(226,12),(233,1),(239,0), (246,1),(251,5),(257,9),(263,0),(266,0),(270,5),(280,8),(285,12),(291,0),(295,5), (305,12),(308,5),(311,1),(315,13),(321,12),(323,1),(327,13),(338,1),(343,5),(347,5), (356,13),(363,5),(369,5),(373,13),(376,1),(382,1),(389,13),(392,5),(394,8),(397,5)}
T′₃={(2,4),(8,9),(12,8),(14,8),(23,12),(29,9),(33,9),(45,0),(47,8),(60,0), (73,13),(78,12),(89,12),(99,5),(102,9),(111,5),(117,4),(127,12),(133,12),(136,1), (138,12),(143,4),(157,12),(160,0),(163,8),(170,12),(176,4),(178,8),(182,8),(187,8), (189,8),(192,1),(198,8),(200,0),(204,4),(210,1),(217,2),(224,0),(228,9),(236,8), (241,0),(249,1),(256,8),(259,9),(264,1),(267,1),(271,9),(282,0),(286,1),(292,8), (302,12),(306,9),(309,4),(312,13),(316,1),(322,5),(324,5),(328,5),(342,12),(346,13), (349,0),(361,1),(367,5),(372,5),(374,4),(380,4),(387,5),(390,5),(393,1),(395,9)}
T′₄={(2,14),(5,3),(12,11),(15,0),(20,13),(22,11),(36,13),(39,5),(43,0), (46,15),(48,13),(53,14),(58,8),(61,11),(71,10),(74,14),(85,7),(88,5),(90,2),(95,6), (103,4),(105,6),(115,8),(130,12),(136,2),(142,11),(155,0),(160,7),(171,10),(182,12), (185,12),(191,6),(199,10),(205,8),(213,1),(224,0),(228,0),(233,0),(236,1),(239,1), (246,13),(248,1),(250,4),(253,9),(258,5),(264,7),(270,4),(274,8),(280,2),(287,1), (294,12),(297,5),(301,7),(304,11),(308,11),(314,15),(316,14),(322,10),(326,8), (331,11),(334,5),(345,0),(351,9),(360,1),(363,3),(375,9),(378,2),(387,1),(394,13), (399,13)}
T′₅={(1,2),(3,3),(6,10),(13,3),(19,13),(21,6),(33,10),(38,6),(40,12),(44,3), (47,3),(51,7),(57,15),(59,6),(70,7),(73,10),(79,1),(86,14),(89,5),(92,10),(99,14), (104,6),(111,10),(128,3),(133,4),(139,6),(145,14),(156,14),(168,5),(176,12),(183,8), (186,9),(193,15),(204,0),(207,5),(217,4),(226,0),(230,4),(234,12),(238,8),(240,3), (247,12),(249,8),(251,1),(254,0),(262,3),(268,6),(271,1),(277,5),(286,11),(293,3), (296,7),(299,7),(302,11),(306,3),(309,15),(315,8),(317,10),(325,11),(327,15), (333,1),(335,9),(348,13),(358,9),(362,15),(371,3),(376,7),(379,1),(389,11), (396,15)}
T′₆={(3,15),(7,13),(14,6),(18,3),(29,12),(32,4),(40,5),(53,1),(55,8),(64,15), (70,5),(75,14),(81,5),(90,8),(95,9),(100,8),(103,0),(105,15),(108,13),(110,5), (113,9),(121,12),(124,13),(132,11),(136,12),(140,9),(145,9),(152,6),(159,5), (164,12),(167,1),(171,12),(176,0),(181,6),(183,10),(188,1),(190,9),(199,15), (209,11),(215,6),(232,0),(235,2),(244,2),(248,0),(254,2),(256,0),(260,7),(266,4), (278,0),(281,2),(288,10),(292,10),(295,4),(299,15),(304,2),(314,7),(322,11),(327,0), (336,6),(346,3),(351,9),(353,1),(364,10),(369,15),(371,15),(374,6),(377,14),(386,3), (389,7),(399,10)}
T′₇={(2,4),(4,7),(9,8),(15,5),(22,4),(30,7),(36,14),(50,5),(54,11),(60,8), (68,12),(71,9),(77,2),(85,10),(91,11),(96,11),(101,5),(104,4),(107,13),(109,13), (111,9),(116,0),(123,12),(131,9),(133,1),(137,11),(144,12),(149,0),(155,7),(162,0), (166,6),(168,9),(174,5),(180,5),(182,0),(184,11),(189,1),(197,6),(203,6),(213,15), (223,3),(234,6),(240,10),(247,6),(253,6),(255,2),(257,10),(265,2),(274,0),(280,4), (286,2),(291,2),(294,1),(297,0),(301,2),(306,3),(321,1),(325,10),(334,11),(338,8), (348,3),(352,8),(355,11),(366,10),(370,14),(373,13),(376,15),(382,5),(387,2), (396,8)}
T′₈={(1,5),(16,9),(18,0),(21,1),(28,9),(31,1),(36,1),(39,0),(44,13),(55,0), (58,7),(72,11),(74,11),(78,3),(81,2),(87,8),(93,7),(99,7),(108,2),(114,3),(118,10), (120,15),(125,11),(132,0),(140,3),(147,2),(155,13),(167,6),(173,11),(175,10), (181,4),(188,10),(195,10),(200,3),(206,15),(208,12),(212,7),(217,0),(222,4), (228,12),(231,12),(242,14),(253,6),(255,8),(257,6),(261,7),(264,7),(268,8),(275,15), (278,15),(285,3),(289,11),(295,0),(298,4),(307,11),(315,1),(318,5),(324,5),(338,8), (342,13),(347,13),(356,4),(358,13),(360,5),(363,1),(365,0),(371,3),(385,1),(389,0), (398,11)}
T′₉={(0,0),(5,9),(17,1),(19,0),(26,0),(29,8),(34,7),(37,5),(40,15),(50,9), (56,0),(67,15),(73,1),(76,5),(80,3),(84,0),(89,5),(96,2),(107,14),(110,0),(117,1), (119,10),(123,11),(128,10),(138,10),(143,2),(152,14),(164,13),(171,2),(174,8), (180,7),(182,14),(191,2),(199,9),(204,14),(207,0),(211,6),(213,7),(218,15),(223,2), (230,9),(241,6),(249,15),(254,1),(256,1),(259,11),(262,12),(265,15),(269,12), (277,11),(280,11),(286,13),(293,13),(297,3),(300,12),(311,1),(316,6),(320,4), (326,9),(339,13),(346,5),(348,7),(357,2),(359,2),(361,0),(364,8),(368,13),(375,8), (388,3),(393,5)}
T′₁₀={(6,9),(12,3),(19,14),(24,15),(31,14),(35,10),(38,6),(41,9),(44,4), (49,7),(59,5),(65,0),(76,10),(84,5),(88,12),(105,14),(109,6),(114,14),(117,3), (121,8),(124,4),(131,14),(135,9),(137,10),(141,8),(144,0),(151,7),(159,12),(169,15), (176,0),(184,1),(193,13),(195,4),(197,4),(209,15),(214,5),(218,14),(222,3),(225,5), (228,9),(232,5),(236,3),(243,8),(247,4),(250,0),(263,8),(267,1),(274,10),(278,15), (282,8),(288,14),(294,10),(300,11),(308,2),(313,8),(319,0),(327,1),(332,4),(335,2), (338,11),(351,4),(354,5),(356,10),(360,1),(363,2),(368,14),(376,10),(379,6),(385,7), (398,15)}
T′₁₁={(1,14),(8,11),(18,10),(21,9),(25,10),(34,11),(37,8),(40,7),(43,3), (45,14),(53,7),(61,13),(72,11),(83,15),(86,11),(101,5),(108,5),(113,6),(115,13), (119,0),(122,7),(129,1),(132,1),(136,8),(139,3),(142,15),(148,12),(154,11),(166,1), (175,3),(183,13),(186,3),(194,3),(196,3),(205,4),(210,0),(217,10),(219,4),(223,0), (227,9),(229,9),(235,5),(241,1),(245,0),(248,5),(262,8),(266,5),(272,1),(275,1), (281,0),(287,15),(291,14),(298,1),(303,10),(309,10),(317,0),(325,2),(329,2),(333,2), (336,0),(350,10),(353,3),(355,5),(359,6),(362,15),(366,2),(373,6),(378,12),(383,1), (388,14)}
T′₁₂={(6,3),(11,15),(15,3),(22,5),(28,12),(33,1),(36,13),(42,8),(44,1), (55,11),(69,9),(71,9),(75,9),(78,10),(81,1),(86,10),(88,13),(95,11),(102,1),(104,9), (110,4),(112,9),(117,2),(120,1),(130,1),(134,4),(138,9),(147,0),(150,11),(155,10), (168,7),(171,3),(180,11),(188,0),(196,3),(201,10),(205,7),(208,13),(215,2),(217,8), (227,14),(231,5),(234,0),(241,6),(248,0),(261,3),(263,5),(266,1),(269,6),(279,2), (287,15),(290,6),(294,1),(296,4),(302,11),(313,10),(321,14),(326,0),(336,4),(342,7), (355,13),(358,11),(369,9),(371,3),(385,2),(388,2),(391,8),(395,2),(397,5),(399,3)}
T′₁₃={(4,8),(7,10),(13,11),(16,11),(27,3),(31,12),(34,13),(38,13),(43,12), (54,14),(57,12),(70,1),(73,14),(76,8),(80,8),(84,9),(87,13),(93,0),(101,0),(103,1), (107,1),(111,1),(114,9),(119,1),(122,8),(131,1),(135,10),(139,15),(149,3),(153,2), (165,13),(170,11),(173,8),(185,3),(193,10),(198,1),(203,11),(207,14),(211,4), (216,7),(220,3),(229,5),(233,14),(237,6),(242,6),(258,6),(262,14),(264,2),(268,12), (273,14),(283,14),(288,0),(292,2),(295,10),(300,4),(308,12),(317,6),(325,1), (332,10),(341,10),(343,7),(356,1),(365,7),(370,3),(377,2),(387,4),(390,7),(392,5), (396,1),(398,7)}
T′₁₄={(1,1),(4,8),(15,15),(18,14),(22,8),(27,14),(33,10),(38,14),(45,11), (52,14),(59,3),(63,14),(66,14),(74,6),(80,2),(84,0),(99,8),(107,13),(110,1),(121,2), (125,3),(131,7),(137,10),(139,1),(145,7),(147,5),(151,2),(154,10),(165,3),(173,11), (179,9),(184,3),(186,2),(189,6),(196,4),(204,3),(206,5),(215,6),(222,1),(225,1), (229,3),(232,13),(237,2),(242,10),(245,5),(248,9),(251,3),(255,9),(259,1),(263,0), (268,1),(270,9),(276,9),(285,9),(289,11),(292,5),(297,1),(308,1),(314,0),(318,12), (322,12),(328,9),(339,13),(341,0),(347,15),(360,10),(368,5),(375,3),(382,4), (396,11)}
T′₁₅={(0,15),(3,6),(10,9),(17,10),(20,9),(25,14),(31,4),(36,3),(41,10),(48,0), (55,10),(62,12),(64,3),(69,0),(77,5),(82,4),(87,6),(103,2),(108,10),(118,12),(122,7), (129,6),(132,7),(138,6),(142,15),(146,3),(148,11),(153,9),(157,0),(168,0),(175,15), (183,8),(185,1),(187,2),(194,9),(203,11),(205,11),(209,1),(219,1),(224,0),(226,1), (231,7),(234,0),(238,2),(243,11),(247,1),(250,2),(253,0),(256,4),(262,9),(267,3), (269,0),(271,8),(281,9),(286,10),(290,8),(293,10),(306,0),(309,1),(317,3),(320,13), (324,12),(336,12),(340,7),(346,15),(348,11),(367,0),(372,11),(378,6),(390,6)}
T′₁₆={(3,13),(11,9),(24,3),(26,4),(31,4),(34,2),(44,0),(51,8),(54,8),(57,2), (60,1),(64,2),(67,1),(69,5),(72,7),(83,12),(95,13),(98,4),(100,8),(103,8),(108,8), (118,4),(126,11),(130,2),(138,9),(145,0),(149,6),(152,14),(158,14),(162,2),(166,1), (170,10),(178,8),(181,2),(190,0),(196,14),(202,1),(205,0),(209,2),(214,3),(231,9), (239,10),(242,6),(244,14),(247,9),(254,8),(268,7),(277,8),(284,10),(288,7),(295,0), (305,14),(308,10),(312,6),(317,14),(323,11),(328,6),(330,11),(340,13),(345,5), (347,6),(352,7),(355,4),(361,14),(367,3),(372,9),(377,7),(382,6),(389,9),(394,7)}
T′₁₇={(2,8),(8,11),(16,14),(25,1),(28,12),(33,5),(39,4),(50,3),(52,14),(56,7), (58,8),(63,12),(66,3),(68,12),(70,7),(80,9),(90,9),(97,3),(99,12),(101,2),(105,2), (109,5),(119,5),(127,5),(133,0),(140,4),(147,1),(151,6),(156,1),(161,4),(163,1), (169,1),(177,2),(179,0),(186,4),(193,3),(201,2),(203,0),(207,11),(211,2),(227,3), (233,9),(241,2),(243,8),(245,13),(251,0),(257,0),(276,10),(282,10),(285,3),(292,4), (304,12),(307,13),(310,7),(314,12),(318,13),(324,11),(329,10),(335,11),(344,14), (346,7),(350,13),(354,3),(357,3),(365,5),(370,12),(376,4),(380,12),(386,2),(392,0)}
T′₁₈={(3,11),(5,15),(10,0),(21,2),(26,1),(33,10),(46,2),(54,8),(62,5),(70,0), (73,9),(80,10),(82,10),(86,6),(91,4),(96,10),(99,2),(111,1),(116,5),(123,10),(138,8), (141,7),(146,14),(152,4),(157,3),(159,13),(167,0),(171,4),(174,1),(177,9),(181,4), (190,9),(196,0),(199,0),(201,0),(204,14),(208,5),(220,8),(229,15),(233,6),(237,8), (247,1),(252,10),(257,7),(263,2),(265,11),(269,0),(271,11),(275,10),(286,2), (296,10),(302,2),(311,3),(316,8),(320,2),(331,13),(333,2),(339,9),(344,6),(346,6), (351,10),(363,4),(367,4),(369,2),(375,1),(378,11),(384,11),(390,10),(394,1), (397,11)}
T′₁₉={(1,2),(4,1),(9,7),(20,2),(25,8),(32,6),(37,4),(53,7),(58,3),(68,0), (72,0),(74,3),(81,2),(84,13),(89,0),(92,5),(97,5),(102,13),(115,5),(118,4),(132,6), (139,2),(143,7),(149,2),(154,4),(158,1),(162,0),(170,6),(172,4),(175,1),(178,14), (184,12),(195,8),(198,12),(200,0),(202,12),(207,3),(216,15),(226,0),(230,12), (235,8),(246,15),(248,7),(255,12),(258,3),(264,14),(267,2),(270,8),(272,8),(279,6), (293,0),(298,14),(307,7),(313,15),(319,14),(330,10),(332,8),(335,6),(340,2), (345,10),(350,10),(362,14),(366,14),(368,1),(373,0),(377,3),(380,0),(387,4),(392,3), (395,0)}
T′₂₀={(3,2),(6,11),(9,10),(11,4),(16,14),(18,2),(31,6),(38,2),(43,0),(49,0), (51,12),(53,6),(60,2),(62,6),(69,8),(79,2),(88,0),(92,8),(97,0),(101,14),(108,10), (112,2),(116,10),(124,14),(129,8),(133,10),(143,0),(145,8),(151,10),(157,0), (159,10),(165,8),(171,8),(174,2),(184,2),(192,14),(204,0),(209,2),(219,6),(222,2), (228,2),(237,2),(244,12),(252,14),(255,8),(258,14),(262,8),(273,14),(279,8),(292,2), (299,10),(303,14),(315,0),(320,6),(324,2),(330,4),(332,0),(336,8),(339,0),(343,4), (349,0),(358,8),(367,14),(376,4),(378,12),(381,10),(388,0),(391,8),(394,0),(399,4)}
T′₂₁={(0,10),(4,6),(8,0),(10,0),(13,12),(17,0),(30,10),(34,8),(42,7),(45,2), (50,8),(52,6),(58,10),(61,8),(67,0),(74,2),(87,4),(91,6),(93,2),(98,10),(104,2), (111,10),(115,2),(122,2),(126,8),(130,0),(135,2),(144,8),(150,8),(156,10),(158,10), (161,0),(170,2),(173,4),(182,14),(191,14),(198,4),(206,15),(216,12),(220,6),(226,6), (232,8),(240,14),(245,0),(253,10),(257,10),(259,4),(267,2),(278,6),(290,10),(296,6), (301,2),(310,6),(319,6),(323,8),(326,6),(331,4),(333,8),(337,10),(340,0),(344,4), (352,8),(366,8),(372,0),(377,14),(379,4),(383,12),(389,8),(393,6),(397,12)}
T′₂₂={(6,8),(13,10),(16,2),(20,10),(30,0),(37,2),(42,14),(48,12),(51,7), (57,11),(61,1),(73,3),(78,11),(83,2),(90,10),(93,11),(96,15),(98,5),(108,0),(113,4), (116,1),(126,7),(131,3),(138,1),(144,5),(147,11),(149,1),(151,3),(154,3),(164,9), (167,3),(169,7),(178,3),(181,5),(186,5),(193,3),(198,5),(203,13),(212,9),(215,1), (218,15),(220,3),(225,1),(227,6),(230,8),(233,2),(238,2),(244,6),(249,11),(256,1), (265,2),(276,8),(280,0),(286,1),(296,2),(303,2),(311,0),(317,0),(319,0),(327,0), (336,0),(348,8),(358,8),(360,8),(367,10),(373,0),(375,8),(390,10),(394,2),(398,10)}
T′₂₃={(4,2),(10,10),(15,10),(18,10),(23,6),(32,10),(39,14),(45,0),(49,2), (52,6),(59,11),(66,0),(75,14),(82,3),(87,13),(92,11),(95,1),(97,3),(106,7),(109,3), (115,7),(119,1),(130,14),(135,5),(142,7),(146,9),(148,0),(150,3),(153,3),(159,1), (166,11),(168,3),(170,9),(180,1),(183,1),(192,3),(197,3),(202,3),(210,9),(213,3), (217,1),(219,5),(221,1),(226,3),(229,12),(231,2),(235,10),(239,1),(246,5),(250,9), (262,7),(275,2),(279,14),(283,8),(291,0),(300,2),(309,0),(312,0),(318,12),(323,0), (329,0),(345,8),(354,8),(359,12),(364,0),(371,4),(374,0),(388,1),(391,10),(397,0)}
T′₂₄={(7,3),(11,13),(16,1),(23,0),(27,8),(30,11),(45,7),(48,7),(59,1),(63,3), (65,7),(70,2),(80,3),(83,7),(85,11),(94,11),(98,9),(101,1),(109,0),(113,9),(117,5), (121,1),(124,3),(128,1),(140,9),(142,5),(160,5),(163,9),(167,11),(176,1),(178,1), (180,2),(185,0),(189,0),(192,15),(202,8),(208,10),(219,2),(224,3),(227,6),(239,3), (241,0),(243,6),(250,4),(252,8),(256,0),(260,1),(265,0),(272,0),(277,2),(284,4), (290,2),(301,10),(311,2),(313,12),(321,10),(323,12),(325,0),(331,2),(335,10), (337,10),(344,10),(349,6),(353,8),(358,2),(360,1),(380,4),(384,11),(392,2),(397,3)}
T′₂₅={(1,3),(8,1),(14,2),(22,11),(24,15),(28,9),(44,9),(46,3),(50,3),(62,3), (64,3),(66,7),(76,1),(82,5),(84,9),(86,5),(97,5),(100,15),(102,3),(112,9),(114,7), (120,1),(123,1),(127,11),(134,3),(141,9),(152,1),(162,3),(165,1),(172,5),(177,0), (179,6),(181,1),(187,7),(190,2),(198,6),(206,2),(218,0),(221,4),(225,0),(233,6), (240,2),(242,0),(249,4),(251,8),(254,8),(259,1),(263,4),(270,1),(273,4),(278,2), (287,2),(293,6),(307,10),(312,6),(316,10),(322,12),(324,0),(330,8),(334,10),(336,8), (341,14),(347,10),(350,4),(355,0),(359,12),(369,2),(383,14),(391,3),(393,13)}
T′₂₆={(7,0),(11,11),(13,1),(23,3),(25,3),(31,9),(35,9),(41,11),(43,3),(48,7), (51,13),(53,3),(65,15),(71,1),(85,9),(88,7),(96,3),(102,7),(106,7),(114,2),(120,2), (123,3),(127,1),(130,4),(142,11),(146,3),(148,15),(154,0),(157,9),(162,6),(166,14), (175,8),(179,11),(190,9),(193,4),(199,0),(206,1),(209,0),(212,2),(214,0),(218,1), (222,0),(224,10),(230,0),(236,2),(246,0),(261,10),(271,0),(276,10),(282,0),(284,8), (290,0),(298,10),(304,8),(307,4),(314,4),(316,2),(319,12),(325,12),(334,1),(342,4), (350,1),(352,2),(354,11),(364,13),(368,9),(372,0),(382,4),(385,1),(399,4)}
T′₂₇={(5,3),(9,1),(12,3),(15,1),(24,3),(27,1),(33,9),(36,9),(42,3),(47,1), (49,11),(52,1),(58,5),(69,13),(81,3),(86,9),(94,11),(97,2),(104,15),(113,10),(115,3), (122,1),(125,5),(129,1),(140,3),(144,1),(147,5),(152,3),(156,2),(160,10),(165,12), (173,8),(176,2),(183,6),(192,10),(195,7),(205,2),(208,6),(210,10),(213,8),(216,10), (221,10),(223,8),(229,10),(231,2),(243,9),(249,13),(266,4),(273,6),(281,6),(283,0), (289,12),(293,2),(300,0),(305,8),(312,0),(315,0),(318,10),(320,8),(329,0),(338,3), (347,9),(351,2),(353,0),(361,9),(365,11),(371,3),(380,3),(384,0),(386,1)}
T′₂₈={(6,0),(19,2),(25,8),(29,0),(32,0),(40,8),(47,0),(51,2),(54,8),(58,2), (63,0),(66,3),(72,8),(77,0),(79,5),(82,12),(90,1),(92,11),(95,0),(105,10),(109,11), (112,15),(118,9),(124,13),(126,15),(133,11),(140,11),(143,9),(146,11),(155,1), (163,7),(172,1),(178,1),(184,8),(187,11),(197,11),(206,3),(211,3),(219,1),(225,11), (237,9),(240,3),(252,9),(264,3),(274,9),(276,5),(280,9),(287,1),(293,9),(298,2), (301,1),(313,0),(316,12),(319,12),(321,10),(327,1),(330,0),(334,2),(339,0),(359,1), (365,5),(370,10),(372,2),(379,1),(382,8),(386,4),(388,4),(393,0),(396,2),(399,0)}
T′₂₉={(0,8),(12,0),(20,0),(26,10),(30,2),(34,0),(41,0),(49,0),(52,2),(55,0), (60,0),(64,0),(69,1),(75,9),(78,0),(81,9),(87,4),(91,15),(94,1),(100,11),(106,10), (110,2),(115,1),(120,3),(125,0),(132,3),(135,3),(141,2),(145,9),(153,7),(161,11), (167,8),(175,15),(183,0),(186,1),(194,3),(201,3),(208,13),(216,3),(223,3),(235,9), (238,3),(246,11),(259,1),(273,2),(275,9),(277,11),(282,9),(289,5),(294,0),(300,3), (310,1),(315,10),(318,0),(320,8),(322,1),(328,2),(331,0),(337,4),(340,2),(362,8), (368,8),(371,1),(374,2),(381,8),(384,4),(387,0),(391,3),(395,2),(398,10)}
T′₃₀={(4,10),(11,0),(16,0),(26,2),(28,2),(32,3),(38,2),(42,1),(54,10),(59,0), (63,1),(67,11),(71,3),(74,10),(82,0),(87,1),(94,8),(106,9),(116,8),(121,11),(126,9), (131,9),(134,1),(136,1),(140,1),(149,1),(156,1),(158,3),(162,11),(165,9),(172,0), (174,3),(181,1),(185,1),(191,3),(195,9),(201,1),(208,1),(212,8),(215,11),(219,8), (227,11),(232,3),(243,9),(252,2),(260,0),(265,9),(272,1),(281,10),(283,9),(288,10), (291,11),(303,3),(310,2),(318,11),(323,3),(330,1),(337,8),(339,2),(342,0),(344,0), (349,10),(353,8),(359,0),(366,8),(374,10),(379,10),(383,8),(390,2),(395,0)}
T′₃₁={(2,2),(8,8),(13,0),(21,8),(27,10),(29,0),(36,0),(39,11),(51,2),(56,0), (61,0),(65,3),(68,0),(72,3),(79,1),(84,8),(88,3),(102,1),(112,11),(118,0),(124,3), (127,1),(132,0),(135,1),(139,1),(142,1),(150,1),(157,1),(159,1),(163,1),(166,2), (173,1),(179,1),(184,1),(189,1),(194,9),(197,0),(203,3),(209,8),(214,3),(218,11), (221,1),(230,3),(234,3),(244,11),(253,1),(261,11),(271,11),(273,1),(282,2),(285,8), (289,8),(294,2),(306,10),(317,2),(319,9),(329,1),(333,2),(338,2),(341,10),(343,2), (345,0),(350,0),(356,0),(362,3),(370,0),(377,0),(381,8),(384,0),(391,8)}
T′₃₂={(11,5),(19,6),(27,6),(30,7),(42,6),(54,4),(56,2),(64,6),(67,2),(72,0), (75,12),(77,4),(80,0),(89,6),(93,6),(96,5),(100,0),(113,14),(118,2),(126,12),(133,6), (137,14),(144,14),(148,2),(151,0),(155,2),(162,2),(164,2),(169,3),(182,12),(187,3), (190,1),(197,1),(210,12),(212,0),(218,3),(224,4),(228,10),(235,8),(237,1),(244,7), (252,0),(258,3),(261,1),(266,1),(270,1),(276,2),(279,0),(284,7),(288,1),(295,7), (298,1),(300,3),(302,1),(304,3),(308,1),(313,1),(321,5),(326,5),(331,10),(334,1), (337,5),(344,4),(356,5),(358,4),(366,5),(374,4),(385,1),(395,4),(398,6)}
T′₃₃={(7,4),(17,3),(22,5),(29,6),(35,14),(47,14),(55,4),(63,1),(66,0),(71,13), (73,0),(76,10),(79,0),(85,7),(90,4),(94,3),(98,4),(106,2),(116,4),(122,2),(128,6), (136,10),(141,2),(147,4),(150,0),(153,14),(157,2),(163,7),(167,1),(176,1),(185,4), (189,0),(191,11),(203,5),(211,0),(216,0),(220,11),(225,7),(232,3),(236,0),(238,9), (245,1),(256,1),(260,7),(265,7),(269,5),(275,3),(277,5),(280,7),(285,9),(290,1), (297,9),(299,3),(301,0),(303,1),(305,1),(311,0),(315,1),(325,2),(328,5),(332,13), (335,7),(341,5),(355,4),(357,5),(362,6),(373,4),(381,1),(386,13),(397,5)}
T′₃₄={(2,5),(12,1),(19,1),(22,5),(24,3),(28,9),(35,13),(39,1),(41,0),(44,10), (47,0),(57,1),(64,5),(68,5),(79,1),(85,1),(91,1),(94,4),(103,3),(105,4),(107,5), (112,7),(119,5),(126,1),(143,2),(145,12),(153,5),(160,4),(168,0),(172,0),(178,4), (188,0),(192,8),(195,4),(200,0),(213,11),(229,1),(231,0),(242,4),(244,4),(247,4), (252,4),(254,4),(264,0),(269,7),(272,0),(281,2),(284,0),(289,2),(299,10),(305,3), (307,7),(320,14),(322,7),(328,6),(331,10),(343,0),(346,0),(352,11),(354,2),(359,1), (363,3),(365,6),(369,5),(378,5),(382,7),(384,12),(386,0),(393,3),(399,1)}
T′₃₅={(0,1),(7,2),(14,5),(21,5),(23,7),(27,2),(32,1),(37,1),(40,1),(43,1), (46,1),(53,5),(62,11),(65,5),(77,7),(83,2),(86,4),(92,3),(95,4),(104,0),(106,8), (109,7),(113,2),(120,7),(135,1),(144,4),(146,13),(154,5),(164,6),(169,2),(177,13), (179,7),(190,4),(194,8),(196,0),(201,5),(215,4),(230,4),(235,0),(243,0),(246,6), (249,4),(253,4),(258,0),(266,4),(271,0),(274,4),(283,0),(287,6),(298,7),(304,1), (306,4),(309,6),(321,1),(327,7),(329,2),(333,6),(345,3),(351,1),(353,5),(357,2), (362,0),(364,7),(368,5),(374,0),(381,2),(383,3),(385,6),(390,3),(396,1)}
T′₃₆={(9,3),(14,2),(23,7),(26,2),(29,12),(39,4),(43,3),(48,3),(50,0),(61,2), (65,0),(67,14),(69,4),(74,4),(76,2),(89,6),(93,6),(100,4),(106,1),(122,4),(124,14), (128,0),(130,4),(136,4),(143,5),(148,4),(154,0),(161,4),(170,4),(174,1),(182,5), (188,6),(192,1),(202,0),(211,9),(216,1),(221,1),(223,3),(234,5),(239,5),(255,5), (260,13),(267,7),(272,5),(277,3),(279,1),(283,6),(287,0),(294,1),(297,1),(304,0), (307,5),(313,2),(323,3),(328,2),(332,1),(341,0),(343,3),(348,1),(352,1),(354,1), (357,0),(361,13),(366,0),(375,0),(380,1),(385,0),(387,4),(393,7),(398,5)}
T′₃₇={(0,2),(10,6),(17,6),(24,7),(28,5),(35,5),(41,5),(47,7),(49,4),(60,2), (62,4),(66,5),(68,2),(70,4),(75,6),(79,2),(91,4),(98,7),(103,4),(114,0),(123,6), (125,14),(129,4),(134,6),(137,4),(146,5),(150,5),(160,4),(164,0),(173,0),(177,4), (187,1),(191,0),(200,7),(210,3),(215,1),(220,0),(222,5),(225,0),(238,0),(254,0), (257,1),(261,7),(268,1),(274,3),(278,5),(281,1),(285,3),(290,1),(296,1),(303,1), (305,5),(310,0),(314,1),(326,1),(329,0),(337,0),(342,3),(344,1),(349,1),(353,2), (356,9),(360,1),(363,1),(370,5),(378,5),(381,4),(386,5),(392,6),(396,5)}
T′₃₈={(5,12),(9,8),(14,8),(18,10),(21,9),(25,5),(35,12),(40,12),(42,0),(49,1), (52,12),(56,0),(59,14),(61,10),(68,0),(78,6),(89,6),(96,0),(101,8),(105,1),(114,5), (119,1),(123,5),(129,0),(137,9),(141,1),(149,4),(153,4),(158,11),(161,11),(169,12), (179,11),(188,9),(194,5),(198,1),(201,13),(206,13),(211,1),(214,11),(221,2),(234,1), (237,8),(239,3),(242,7),(248,13),(251,7),(259,3),(261,14),(269,5),(274,0),(282,6), (284,0),(295,3),(299,5),(302,2),(305,4),(310,9),(314,13),(332,12),(339,7),(341,14), (348,12),(352,12),(355,1),(361,0),(370,13),(379,12),(381,11),(392,10),(395,8)}
T′₃₉={(0,4),(8,4),(10,12),(17,0),(20,4),(23,13),(32,9),(37,0),(41,8),(46,1), (50,13),(55,9),(57,6),(60,4),(67,0),(76,4),(83,0),(91,8),(98,0),(102,6),(110,2), (117,15),(121,10),(127,15),(134,7),(139,10),(148,13),(151,9),(156,10),(160,1), (166,1),(172,2),(187,6),(189,2),(197,0),(200,7),(205,2),(210,9),(212,3),(220,1), (222,1),(236,7),(238,1),(240,1),(245,13),(250,5),(258,5),(260,7),(263,13),(270,9), (279,5),(283,4),(291,5),(296,11),(301,6),(303,5),(306,5),(312,4),(324,5),(333,6), (340,13),(345,13),(349,0),(354,0),(357,8),(364,13),(372,14),(380,4),(383,5), (394,10)} (30)
T′₀={(3,10),(7,12),(14,12),(20,9),(35,1),(46,0),(57,3),(71,12),(78,2),(85,9), (92,4),(99,15),(110,13),(117,7),(125,15),(129,9),(150,12),(158,9),(163,5),(168,11), (175,12),(185,9),(195,9),(202,9),(212,13),(217,3),(226,10),(232,5),(241,7),(251,5), (267,4),(273,4),(278,5),(289,6),(297,12),(309,5),(312,5),(334,9),(338,6),(343,1), (350,1),(361,8),(367,6),(375,12),(379,0),(388,4)}
T′₁={(2,8),(6,8),(12,4),(19,0),(34,8),(45,13),(56,9),(63,8),(77,3),(83,13), (90,15),(97,0),(107,14),(116,9),(121,8),(128,15),(134,12),(156,1),(161,7),(165,13), (172,12),(180,5),(194,8),(200,9),(207,13),(214,13),(224,14),(228,7),(240,7), (250,12),(260,5),(272,4),(276,6),(288,1),(292,4),(302,4),(311,3),(330,10),(337,1), (342,10),(349,9),(357,14),(365,12),(373,4),(377,0),(384,1),(391,3)}
T′₂={(1,8),(5,12),(9,8),(15,0),(26,4),(38,0),(48,13),(62,1),(75,8),(82,13), (88,1),(93,12),(104,5),(111,15),(120,9),(127,14),(131,5),(152,13),(159,13),(164,4), (171,9),(177,13),(193,1),(199,3),(204,3),(213,1),(223,13),(227,5),(236,14),(245,9), (255,5),(268,4),(275,9),(284,8),(291,2),(299,4),(310,4),(326,0),(335,4),(340,14), (347,4),(351,4),(364,0),(369,2),(376,12),(383,0),(389,0)}
T′₃={(8,9),(13,4),(23,12),(30,12),(45,0),(56,4),(73,13),(81,13),(99,5), (107,4),(117,4),(128,5),(136,1),(141,12),(157,12),(161,0),(170,12),(177,4),(182,8), (188,8),(192,1),(199,8),(204,4),(214,4),(224,0),(233,1),(241,0),(251,5),(259,9), (266,0),(271,9),(285,12),(292,8),(305,12),(309,4),(315,13),(322,5),(327,13), (342,12),(347,5),(361,1),(369,5),(374,4),(382,1),(390,5),(394,8)}
T′₄={(5,0),(12,8),(17,1),(29,9),(37,4),(47,8),(65,13),(78,12),(94,5),(102,9), (112,13),(127,12),(134,1),(138,12),(155,0),(160,0),(169,9),(176,4),(180,8),(187,8), (191,8),(198,8),(202,1),(210,1),(221,9),(228,9),(239,0),(249,1),(257,9),(264,1), (270,5),(282,0),(291,0),(302,12),(308,5),(312,13),(321,12),(324,5),(338,1),(346,13), (356,13),(367,5),(373,13),(380,4),(389,13),(393,1),(397,5)}
T′₅={(2,4),(10,8),(14,8),(24,13),(33,9),(46,12),(60,0),(77,13),(89,12), (100,9),(111,5),(125,0),(133,12),(137,0),(143,4),(158,9),(163,8),(174,8),(178,8), (186,12),(189,8),(196,0),(200,0),(207,8),(217,2),(226,12),(236,8),(246,1),(256,8), (263,0),(267,1),(280,8),(286,1),(295,5),(306,9),(311,1),(316,1),(323,1),(328,5), (343,5),(349,0),(363,5),(372,5),(376,1),(387,5),(392,5),(395,9)}
T′₆={(3,3),(12,11),(19,13),(22,11),(38,6),(43,0),(47,3),(53,14),(59,6), (71,10),(79,1),(88,5),(92,10),(103,4),(111,10),(130,12),(139,6),(155,0),(168,5), (182,12),(186,9),(199,10),(207,5),(224,0),(230,4),(236,1),(240,3),(248,1),(251,1), (258,5),(268,6),(274,8),(286,11),(294,12),(299,7),(304,11),(309,15),(316,14), (325,11),(331,11),(335,9),(351,9),(362,15),(375,9),(379,1),(394,13)}
T′₇={(2,14),(6,10),(15,0),(21,6),(36,13),(40,12),(46,15),(51,7),(58,8),(70,7), (74,14),(86,14),(90,2),(99,14),(105,6),(128,3),(136,2),(145,14),(160,7),(176,12), (185,12),(193,15),(205,8),(217,4),(228,0),(234,12),(239,1),(247,12),(250,4),(254,0), (264,7),(271,1),(280,2),(293,3),(297,5),(302,11),(308,11),(315,8),(322,10),(327,15), (334,5),(348,13),(360,1),(371,3),(378,2),(389,11),(399,13)}
T′₈={(1,2),(5,3),(13,3),(20,13),(33,10),(39,5),(44,3),(48,13),(57,15),(61,11), (73,10),(85,7),(89,5),(95,6),(104,6),(115,8),(133,4),(142,11),(156,14),(171,10), (183,8),(191,6),(204,0),(213,1),(226,0),(233,0),(238,8),(246,13),(249,8),(253,9), (262,3),(270,4),(277,5),(287,1),(296,7),(301,7),(306,3),(314,15),(317,10),(326,8), (333,1),(345,0),(358,9),(363,3),(376,7),(387,1),(396,15)}
T′₉={(4,7),(14,6),(22,4),(32,4),(50,5),(55,8),(68,12),(75,14),(85,10),(95,9), (101,5),(105,15),(109,13),(113,9),(123,12),(132,11),(137,11),(145,9),(155,7), (164,12),(168,9),(176,0),(182,0),(188,1),(197,6),(209,11),(223,3),(235,2),(247,6), (254,2),(257,10),(266,4),(280,4),(288,10),(294,1),(299,15),(306,3),(322,11), (334,11),(346,3),(352,8),(364,10),(370,14),(374,6),(382,5),(389,7)}
T′₁₀={(3,15),(9,8),(18,3),(30,7),(40,5),(54,11),(64,15),(71,9),(81,5),(91,11), (100,8),(104,4),(108,13),(111,9),(121,12),(131,9),(136,12),(144,12),(152,6),(162,0), (167,1),(174,5),(181,6),(184,11),(190,9),(203,6),(215,6),(234,6),(244,2),(253,6), (256,0),(265,2),(278,0),(286,2),(292,10),(297,0),(304,2),(321,1),(327,0),(338,8), (351,9),(355,11),(369,15),(373,13),(377,14),(387,2),(399,10)}
T′₁₁={(2,4),(7,13),(15,5),(29,12),(36,14),(53,1),(60,8),(70,5),(77,2),(90,8), (96,11),(103,0),(107,13),(110,5),(116,0),(124,13),(133,1),(140,9),(149,0),(159,5), (166,6),(171,12),(180,5),(183,10),(189,1),(199,15),(213,15),(232,0),(240,10), (248,0),(255,2),(260,7),(274,0),(281,2),(291,2),(295,4),(301,2),(314,7),(325,10), (336,6),(348,3),(353,1),(366,10),(371,15),(376,15),(386,3),(396,8)}
T′₁₂={(5,9),(18,0),(26,0),(31,1),(37,5),(44,13),(56,0),(72,11),(76,5),(81,2), (89,5),(99,7),(110,0),(118,10),(123,11),(132,0),(143,2),(155,13),(171,2),(175,10), (182,14),(195,10),(204,14),(208,12),(213,7),(222,4),(230,9),(242,14),(254,1), (257,6),(262,12),(268,8),(277,11),(285,3),(293,13),(298,4),(311,1),(318,5),(326,9), (342,13),(348,7),(358,13),(361,0),(365,0),(375,8),(389,0)}
T′₁₃={(1,5),(17,1),(21,1),(29,8),(36,1),(40,15),(55,0),(67,15),(74,11),(80,3), (87,8),(96,2),(108,2),(117,1),(120,15),(128,10),(140,3),(152,14),(167,6),(174,8), (181,4),(191,2),(200,3),(207,0),(212,7),(218,15),(228,12),(241,6),(253,6),(256,1), (261,7),(265,15),(275,15),(280,11),(289,11),(297,3),(307,11),(316,6),(324,5), (339,13),(347,13),(357,2),(360,5),(364,8),(371,3),(388,3),(398,11)}
T′₁₄={(0,0),(16,9),(19,0),(28,9),(34,7),(39,0),(50,9),(58,7),(73,1),(78,3), (84,0),(93,7),(107,14),(114,3),(119,10),(125,11),(138,10),(147,2),(164,13),(173,11), (180,7),(188,10),(199,9),(206,15),(211,6),(217,0),(223,2),(231,12),(249,15),(255,8), (259,11),(264,7),(269,12),(278,15),(286,13),(295,0),(300,12),(315,1),(320,4), (338,8),(346,5),(356,4),(359,2),(363,1),(368,13),(385,1),(393,5)}
T′₁₅={(8,11),(19,14),(25,10),(35,10),(40,7),(44,4),(53,7),(65,0),(83,15), (88,12),(108,5),(114,14),(119,0),(124,4),(132,1),(137,10),(142,15),(151,7),(166,1), (176,0),(186,3),(195,4),(205,4),(214,5),(219,4),(225,5),(229,9),(236,3),(245,0), (250,0),(266,5),(274,10),(281,0),(288,14),(298,1),(308,2),(317,0),(327,1),(333,2), (338,11),(353,3),(356,10),(362,15),(368,14),(378,12),(385,7)}
T′₁₆={(6,9),(18,10),(24,15),(34,11),(38,6),(43,3),(49,7),(61,13),(76,10), (86,11),(105,14),(113,6),(117,3),(122,7),(131,14),(136,8),(141,8),(148,12),(159,12), (175,3),(184,1),(194,3),(197,4),(210,0),(218,14),(223,0),(228,9),(235,5),(243,8), (248,5),(263,8),(272,1),(278,15),(287,15),(294,10),(303,10),(313,8),(325,2),(332,4), (336,0),(351,4),(355,5),(360,1),(366,2),(376,10),(383,1),(398,15)}
T′₁₇={(1,14),(12,3),(21,9),(31,14),(37,8),(41,9),(45,14),(59,5),(72,11),(84,5), (101,5),(109,6),(115,13),(121,8),(129,1),(135,9),(139,3),(144,0),(154,11),(169,15), (183,13),(193,13),(196,3),(209,15),(217,10),(222,3),(227,9),(232,5),(241,1),(247,4), (262,8),(267,1),(275,1),(282,8),(291,14),(300,11),(309,10),(319,0),(329,2),(335,2), (350,10),(354,5),(359,6),(363,2),(373,6),(379,6),(388,14)}
T′₁₈={(7,10),(15,3),(27,3),(33,1),(38,13),(44,1),(57,12),(71,9),(76,8),(81,1), (87,13),(95,11),(103,1),(110,4),(114,9),(120,1),(131,1),(138,9),(149,3),(155,10), (170,11),(180,11),(193,10),(201,10),(207,14),(215,2),(220,3),(231,5),(237,6), (248,0),(262,14),(266,1),(273,14),(287,15),(292,2),(296,4),(308,12),(321,14), (332,10),(342,7),(356,1),(369,9),(377,2),(388,2),(392,5),(397,5)}
T′₁₉={(6,3),(13,11),(22,5),(31,12),(36,13),(43,12),(55,11),(70,1),(75,9), (80,8),(86,10),(93,0),(102,1),(107,1),(112,9),(119,1),(130,1),(135,10),(147,0), (153,2),(168,7),(173,8),(188,0),(198,1),(205,7),(211,4),(217,8),(229,5),(234,0), (242,6),(261,3),(264,2),(269,6),(283,14),(290,6),(295,10),(302,11),(317,6),(326,0), (341,10),(355,13),(365,7),(371,3),(387,4),(391,8),(396,1),(399,3)}
T′₂₀={(4,8),(11,15),(16,11),(28,12),(34,13),(42,8),(54,14),(69,9),(73,14), (78,10),(84,9),(88,13),(101,0),(104,9),(111,1),(117,2),(122,8),(134,4),(139,15), (150,11),(165,13),(171,3),(185,3),(196,3),(203,11),(208,13),(216,7),(227,14), (233,14),(241,6),(258,6),(263,5),(268,12),(279,2),(288,0),(294,1),(300,4),(313,10), (325,1),(336,4),(343,7),(358,11),(370,3),(385,2),(390,7),(395,2),(398,7)}
T′₂₁={(3,6),(15,15),(20,9),(27,14),(36,3),(45,11),(55,10),(63,14),(69,0), (80,2),(87,6),(107,13),(118,12),(125,3),(132,7),(139,1),(146,3),(151,2),(157,0), (173,11),(183,8),(186,2),(194,9),(204,3),(209,1),(222,1),(226,1),(232,13),(238,2), (245,5),(250,2),(255,9),(262,9),(268,1),(271,8),(285,9),(290,8),(297,1),(309,1), (318,12),(324,12),(339,13),(346,15),(360,10),(372,11),(382,4)}
T′₂₂={(1,1),(10,9),(18,14),(25,14),(33,10),(41,10),(52,14),(62,12),(66,14), (77,5),(84,0),(103,2),(110,1),(122,7),(131,7),(138,6),(145,7),(148,11),(154,10), (168,0),(179,9),(185,1),(189,6),(203,11),(206,5),(219,1),(225,1),(231,7),(237,2), (243,11),(248,9),(253,0),(259,1),(267,3),(270,9),(281,9),(289,11),(293,10),(308,1), (317,3),(322,12),(336,12),(341,0),(348,11),(368,5),(378,6),(396,11)}
T′₂₃={(0,15),(4,8),(17,10),(22,8),(31,4),(38,14),(48,0),(59,3),(64,3),(74,6), (82,4),(99,8),(108,10),(121,2),(129,6),(137,10),(142,15),(147,5),(153,9),(165,3), (175,15),(184,3),(187,2),(196,4),(205,11),(215,6),(224,0),(229,3),(234,0),(242,10), (247,1),(251,3),(256,4),(263,0),(269,0),(276,9),(286,10),(292,5),(306,0),(314,0), (320,13),(328,9),(340,7),(347,15),(367,0),(375,3),(390,6)}
T′₂₄={(8,11),(24,3),(28,12),(34,2),(50,3),(54,8),(58,8),(64,2),(68,12),(72,7), (90,9),(98,4),(101,2),(108,8),(119,5),(130,2),(140,4),(149,6),(156,1),(162,2), (169,1),(178,8),(186,4),(196,14),(203,0),(209,2),(227,3),(239,10),(243,8),(247,9), (257,0),(277,8),(285,3),(295,0),(307,13),(312,6),(318,13),(328,6),(335,11),(345,5), (350,13),(355,4),(365,5),(372,9),(380,12),(389,9)}
T′₂₅={(3,13),(16,14),(26,4),(33,5),(44,0),(52,14),(57,2),(63,12),(67,1),(70,7), (83,12),(97,3),(100,8),(105,2),(118,4),(127,5),(138,9),(147,1),(152,14),(161,4), (166,1),(177,2),(181,2),(193,3),(202,1),(207,11),(214,3),(233,9),(242,6),(245,13), (254,8),(276,10),(284,10),(292,4),(305,14),(310,7),(317,14),(324,11),(330,11), (344,14),(347,6),(354,3),(361,14),(370,12),(377,7),(386,2),(394,7)}
T′₂₆={(2,8),(11,9),(25,1),(31,4),(39,4),(51,8),(56,7),(60,1),(66,3),(69,5), (80,9),(95,13),(99,12),(103,8),(109,5),(126,11),(133,0),(145,0),(151,6),(158,14), (163,1),(170,10),(179,0),(190,0),(201,2),(205,0),(211,2),(231,9),(241,2),(244,14), (251,0),(268,7),(282,10),(288,7),(304,12),(308,10),(314,12),(323,11),(329,10), (340,13),(346,7),(352,7),(357,3),(367,3),(376,4),(382,6),(392,0)}
T′₂₇={(4,1),(10,0),(25,8),(33,10),(53,7),(62,5),(72,0),(80,10),(84,13),(91,4), (97,5),(111,1),(118,4),(138,8),(143,7),(152,4),(158,1),(167,0),(172,4),(177,9), (184,12),(196,0),(200,0),(204,14),(216,15),(229,15),(235,8),(247,1),(255,12), (263,2),(267,2),(271,11),(279,6),(296,10),(307,7),(316,8),(330,10),(333,2),(340,2), (346,6),(362,14),(367,4),(373,0),(378,11),(387,4),(394,1)}
T′₂₈={(3,11),(9,7),(21,2),(32,6),(46,2),(58,3),(70,0),(74,3),(82,10),(89,0), (96,10),(102,13),(116,5),(132,6),(141,7),(149,2),(157,3),(162,0),(171,4),(175,1), (181,4),(195,8),(199,0),(202,12),(208,5),(226,0),(233,6),(246,15),(252,10),(258,3), (265,11),(270,8),(275,10),(293,0),(302,2),(313,15),(320,2),(332,8),(339,9),(345,10), (351,10),(366,14),(369,2),(377,3),(384,11),(392,3),(397,11)}
T′₂₉={(1,2),(5,15),(20,2),(26,1),(37,4),(54,8),(68,0),(73,9),(81,2),(86,6), (92,5),(99,2),(115,5),(123,10),(139,2),(146,14),(154,4),(159,13),(170,6),(174,1), (178,14),(190,9),(198,12),(201,0),(207,3),(220,8),(230,12),(237,8),(248,7),(257,7), (264,14),(269,0),(272,8),(286,2),(298,14),(311,3),(319,14),(331,13),(335,6),(344,6), (350,10),(363,4),(368,1),(375,1),(380,0),(390,10),(395,0)}
T′₃₀={(4,6),(9,10),(13,12),(18,2),(34,8),(43,0),(50,8),(53,6),(61,8),(69,8), (87,4),(92,8),(98,10),(108,10),(115,2),(124,14),(130,0),(143,0),(150,8),(157,0), (161,0),(171,8),(182,14),(192,14),(206,15),(219,6),(226,6),(237,2),(245,0),(255,8), (259,4),(273,14),(290,10),(299,10),(310,6),(320,6),(326,6),(332,0),(337,10),(343,4), (352,8),(367,14),(377,14),(381,10),(389,8),(394,0)}
T′₃₁={(3,2),(8,0),(11,4),(17,0),(31,6),(42,7),(49,0),(52,6),(60,2),(67,0), (79,2),(91,6),(97,0),(104,2),(112,2),(122,2),(129,8),(135,2),(145,8),(156,10), (159,10),(170,2),(174,2),(191,14),(204,0),(216,12),(222,2),(232,8),(244,12), (253,10),(258,14),(267,2),(279,8),(296,6),(303,14),(319,6),(324,2),(331,4),(336,8), (340,0),(349,0),(366,8),(376,4),(379,4),(388,0),(393,6),(399,4)}
T′₃₂={(0,10),(6,11),(10,0),(16,14),(30,10),(38,2),(45,2),(51,12),(58,10), (62,6),(74,2),(88,0),(93,2),(101,14),(111,10),(116,10),(126,8),(133,10),(144,8), (151,10),(158,10),(165,8),(173,4),(184,2),(198,4),(209,2),(220,6),(228,2),(240,14), (252,14),(257,10),(262,8),(278,6),(292,2),(301,2),(315,0),(323,8),(330,4),(333,8), (339,0),(344,4),(358,8),(372,0),(378,12),(383,12),(391,8),(397,12)}
T′₃₃={(10,10),(16,2),(23,6),(37,2),(45,0),(51,7),(59,11),(73,3),(82,3),(90,10), (95,1),(98,5),(109,3),(116,1),(130,14),(138,1),(146,9),(149,1),(153,3),(164,9), (168,3),(178,3),(183,1),(193,3),(202,3),(212,9),(217,1),(220,3),(226,3),(230,8), (235,10),(244,6),(250,9),(265,2),(279,14),(286,1),(300,2),(311,0),(318,12),(327,0), (345,8),(358,8),(364,0),(373,0),(388,1),(394,2)}
T′₃₄={(6,8),(15,10),(20,10),(32,10),(42,14),(49,2),(57,11),(66,0),(78,11), (87,13),(93,11),(97,3),(108,0),(115,7),(126,7),(135,5),(144,5),(148,0),(151,3), (159,1),(167,3),(170,9),(181,5),(192,3),(198,5),(210,9),(215,1),(219,5),(225,1), (229,12),(233,2),(239,1),(249,11),(262,7),(276,8),(283,8),(296,2),(309,0),(317,0), (323,0),(336,0),(354,8),(360,8),(371,4),(375,8),(391,10),(398,10)}
T′₃₅={(4,2),(13,10),(18,10),(30,0),(39,14),(48,12),(52,6),(61,1),(75,14), (83,2),(92,11),(96,15),(106,7),(113,4),(119,1),(131,3),(142,7),(147,11),(150,3), (154,3),(166,11),(169,7),(180,1),(186,5),(197,3),(203,13),(213,3),(218,15),(221,1), (227,6),(231,2),(238,2),(246,5),(256,1),(275,2),(280,0),(291,0),(303,2),(312,0), (319,0),(329,0),(348,8),(359,12),(367,10),(374,0),(390,10),(397,0)}
T′₃₆={(8,1),(16,1),(24,15),(30,11),(46,3),(59,1),(64,3),(70,2),(82,5),(85,11), (97,5),(101,1),(112,9),(117,5),(123,1),(128,1),(141,9),(160,5),(165,1),(176,1), (179,6),(185,0),(190,2),(202,8),(218,0),(224,3),(233,6),(241,0),(249,4),(252,8), (259,1),(265,0),(273,4),(284,4),(293,6),(311,2),(316,10),(323,12),(330,8),(335,10), (341,14),(349,6),(355,0),(360,1),(383,14),(392,2)}
T′₃₇={(7,3),(14,2),(23,0),(28,9),(45,7),(50,3),(63,3),(66,7),(80,3),(84,9), (94,11),(100,15),(109,0),(114,7),(121,1),(127,11),(140,9),(152,1),(163,9),(172,5), (178,1),(181,1),(189,0),(198,6),(208,10),(221,4),(227,6),(240,2),(243,6),(251,8), (256,0),(263,4),(272,0),(278,2),(290,2),(307,10),(313,12),(322,12),(325,0),(334,10), (337,10),(347,10),(353,8),(359,12),(380,4),(391,3),(397,3)}
T′₃₈={(1,3),(11,13),(22,11),(27,8),(44,9),(48,7),(62,3),(65,7),(76,1),(83,7), (86,5),(98,9),(102,3),(113,9),(120,1),(124,3),(134,3),(142,5),(162,3),(167,11), (177,0),(180,2),(187,7),(192,15),(206,2),(219,2),(225,0),(239,3),(242,0),(250,4), (254,8),(260,1),(270,1),(277,2),(287,2),(301,10),(312,6),(321,10),(324,0),(331,2), (336,8),(344,10),(350,4),(358,2),(369,2),(384,11),(393,13)}
T′₃₉={(9,1),(13,1),(24,3),(31,9),(36,9),(43,3),(49,11),(53,3),(69,13),(85,9), (94,11),(102,7),(113,10),(120,2),(125,5),(130,4),(144,1),(148,15),(156,2),(162,6), (173,8),(179,11),(192,10),(199,0),(208,6),(212,2),(216,10),(222,0),(229,10),(236,2), (249,13),(271,0),(281,6),(284,8),(293,2),(304,8),(312,0),(316,2),(320,8),(334,1), (347,9),(352,2),(361,9),(368,9),(380,3),(385,1)}
T′₄₀={(7,0),(12,3),(23,3),(27,1),(35,9),(42,3),(48,7),(52,1),(65,15),(81,3), (88,7),(97,2),(106,7),(115,3),(123,3),(129,1),(142,11),(147,5),(154,0),(160,10), (166,14),(176,2),(190,9),(195,7),(206,1),(210,10),(214,0),(221,10),(224,10),(231,2), (246,0),(266,4),(276,10),(283,0),(290,0),(300,0),(307,4),(315,0),(319,12),(329,0), (342,4),(351,2),(354,11),(365,11),(372,0),(384,0),(399,4)}
T′₄₁={(5,3),(11,11),(15,1),(25,3),(33,9),(41,11),(47,1),(51,13),(58,5),(71,1), (86,9),(96,3),(104,15),(114,2),(122,1),(127,1),(140,3),(146,3),(152,3),(157,9), (165,12),(175,8),(183,6),(193,4),(205,2),(209,0),(213,8),(218,1),(223,8),(230,0), (243,9),(261,10),(273,6),(282,0),(289,12),(298,10),(305,8),(314,4),(318,10), (325,12),(338,3),(350,1),(353,0),(364,13),(371,3),(382,4),(386,1)}
T′₄₂={(12,0),(25,8),(30,2),(40,8),(49,0),(54,8),(60,0),(66,3),(75,9),(79,5), (87,4),(92,11),(100,11),(109,11),(115,1),(124,13),(132,3),(140,11),(145,9),(155,1), (167,8),(178,1),(186,1),(197,11),(208,13),(219,1),(235,9),(240,3),(259,1),(274,9), (277,11),(287,1),(294,0),(301,1),(315,10),(319,12),(322,1),(330,0),(337,4),(359,1), (368,8),(372,2),(381,8),(386,4),(391,3),(396,2)}
T′₄₃={(6,0),(20,0),(29,0),(34,0),(47,0),(52,2),(58,2),(64,0),(72,8),(78,0), (82,12),(91,15),(95,0),(106,10),(112,15),(120,3),(126,15),(135,3),(143,9),(153,7), (163,7),(175,15),(184,8),(194,3),(206,3),(216,3),(225,11),(238,3),(252,9),(273,2), (276,5),(282,9),(293,9),(300,3),(313,0),(318,0),(321,10),(328,2),(334,2),(340,2), (365,5),(371,1),(379,1),(384,4),(388,4),(395,2),(399,0)}
T′₄₄={(0,8),(19,2),(26,10),(32,0),(41,0),(51,2),(55,0),(63,0),(69,1),(77,0), (81,9),(90,1),(94,1),(105,10),(110,2),(118,9),(125,0),(133,11),(141,2),(146,11), (161,11),(172,1),(183,0),(187,11),(201,3),(211,3),(223,3),(237,9),(246,11),(264,3), (275,9),(280,9),(289,5),(298,2),(310,1),(316,12),(320,8),(327,1),(331,0),(339,0), (362,8),(370,10),(374,2),(382,8),(387,0),(393,0),(398,10)}
T′₄₅={(8,8),(16,0),(27,10),(32,3),(39,11),(54,10),(61,0),(67,11),(72,3),(82,0), (88,3),(106,9),(118,0),(126,9),(132,0),(136,1),(142,1),(156,1),(159,1),(165,9), (173,1),(181,1),(189,1),(195,9),(203,3),(212,8),(218,11),(227,11),(234,3),(252,2), (261,11),(272,1),(282,2),(288,10),(294,2),(310,2),(319,9),(330,1),(338,2),(342,0), (345,0),(353,8),(362,3),(374,10),(381,8),(390,2)}
T′₄₆={(4,10),(13,0),(26,2),(29,0),(38,2),(51,2),(59,0),(65,3),(71,3),(79,1), (87,1),(102,1),(116,8),(124,3),(131,9),(135,1),(140,1),(150,1),(158,3),(163,1), (172,0),(179,1),(185,1),(194,9),(201,1),(209,8),(215,11),(221,1),(232,3),(244,11), (260,0),(271,11),(281,10),(285,8),(291,11),(306,10),(318,11),(329,1),(337,8), (341,10),(344,0),(350,0),(359,0),(370,0),(379,10),(384,0),(395,0)}
T′₄₇={(2,2),(11,0),(21,8),(28,2),(36,0),(42,1),(56,0),(63,1),(68,0),(74,10), (84,8),(94,8),(112,11),(121,11),(127,1),(134,1),(139,1),(149,1),(157,1),(162,11), (166,2),(174,3),(184,1),(191,3),(197,0),(208,1),(214,3),(219,8),(230,3),(243,9), (253,1),(265,9),(273,1),(283,9),(289,8),(303,3),(317,2),(323,3),(333,2),(339,2), (343,2),(349,10),(356,0),(366,8),(377,0),(383,8),(391,8)}
T′₄₈={(17,3),(27,6),(35,14),(54,4),(63,1),(67,2),(73,0),(77,4),(85,7),(93,6), (98,4),(113,14),(122,2),(133,6),(141,2),(148,2),(153,14),(162,2),(167,1),(182,12), (189,0),(197,1),(211,0),(218,3),(225,7),(235,8),(238,9),(252,0),(260,7),(266,1), (275,3),(279,0),(285,9),(295,7),(299,3),(302,1),(305,1),(313,1),(325,2),(331,10), (335,7),(344,4),(357,5),(366,5),(381,1),(395,4)}
T′₄₉={(11,5),(22,5),(30,7),(47,14),(56,2),(66,0),(72,0),(76,10),(80,0),(90,4), (96,5),(106,2),(118,2),(128,6),(137,14),(147,4),(151,0),(157,2),(164,2),(176,1), (187,3),(191,11),(210,12),(216,0),(224,4),(232,3),(237,1),(245,1),(258,3),(265,7), (270,1),(277,5),(284,7),(290,1),(298,1),(301,0),(304,3),(311,0),(321,5),(328,5), (334,1),(341,5),(356,5),(362,6),(374,4),(386,13),(398,6)}
T′₅₀={(7,4),(19,6),(29,6),(42,6),(55,4),(64,6),(71,13),(75,12),(79,0),(89,6), (94,3),(100,0),(116,4),(126,12),(136,10),(144,14),(150,0),(155,2),(163,7),(169,3), (185,4),(190,1),(203,5),(212,0),(220,11),(228,10),(236,0),(244,7),(256,1),(261,1), (269,5),(276,2),(280,7),(288,1),(297,9),(300,3),(303,1),(308,1),(315,1),(326,5), (332,13),(337,5),(355,4),(358,4),(373,4),(385,1),(397,5)}
T′₅₁={(7,2),(19,1),(23,7),(28,9),(37,1),(41,0),(46,1),(57,1),(65,5),(79,1), (86,4),(94,4),(104,0),(107,5),(113,2),(126,1),(144,4),(153,5),(164,6),(172,0), (179,7),(192,8),(196,0),(213,11),(230,4),(242,4),(246,6),(252,4),(258,0),(269,7), (274,4),(284,0),(298,7),(305,3),(309,6),(322,7),(329,2),(343,0),(351,1),(354,2), (362,0),(365,6),(374,0),(382,7),(385,6),(393,3)}
T′₅₂={(2,5),(14,5),(22,5),(27,2),(35,13),(40,1),(44,10),(53,5),(64,5),(77,7), (85,1),(92,3),(103,3),(106,8),(112,7),(120,7),(143,2),(146,13),(160,4),(169,2), (178,4),(190,4),(195,4),(201,5),(229,1),(235,0),(244,4),(249,4),(254,4),(266,4), (272,0),(283,0),(289,2),(304,1),(307,7),(321,1),(328,6),(333,6),(346,0),(353,5), (359,1),(364,7),(369,5),(381,2),(384,12),(390,3),(399,1)}
T′₅₃={(0,1),(12,1),(21,5),(24,3),(32,1),(39,1),(43,1),(47,0),(62,11),(68,5), (83,2),(91,1),(95,4),(105,4),(109,7),(119,5),(135,1),(145,12),(154,5),(168,0), (177,13),(188,0),(194,8),(200,0),(215,4),(231,0),(243,0),(247,4),(253,4),(264,0), (271,0),(281,2),(287,6),(299,10),(306,4),(320,14),(327,7),(331,10),(345,3),(352,11), (357,2),(363,3),(368,5),(378,5),(383,3),(386,0),(396,1)}
T′₅₄={(10,6),(23,7),(28,5),(39,4),(47,7),(50,0),(62,4),(67,14),(70,4),(76,2), (91,4),(100,4),(114,0),(124,14),(129,4),(136,4),(146,5),(154,0),(164,0),(174,1), (187,1),(192,1),(210,3),(216,1),(222,5),(234,5),(254,0),(260,13),(268,1),(277,3), (281,1),(287,0),(296,1),(304,0),(310,0),(323,3),(329,0),(341,0),(344,1),(352,1), (356,9),(361,13),(370,5),(380,1),(386,5),(393,7)}
T′₅₅={(9,3),(17,6),(26,2),(35,5),(43,3),(49,4),(61,2),(66,5),(69,4),(75,6), (89,6),(98,7),(106,1),(123,6),(128,0),(134,6),(143,5),(150,5),(161,4),(173,0), (182,5),(191,0),(202,0),(215,1),(221,1),(225,0),(239,5),(257,1),(267,7),(274,3), (279,1),(285,3),(294,1),(303,1),(307,5),(314,1),(328,2),(337,0),(343,3),(349,1), (354,1),(360,1),(366,0),(378,5),(385,0),(392,6),(398,5)}
T′₅₆={(0,2),(14,2),(24,7),(29,12),(41,5),(48,3),(60,2),(65,0),(68,2),(74,4), (79,2),(93,6),(103,4),(122,4),(125,14),(130,4),(137,4),(148,4),(160,4),(170,4), (177,4),(188,6),(200,7),(211,9),(220,0),(223,3),(238,0),(255,5),(261,7),(272,5), (278,5),(283,6),(290,1),(297,1),(305,5),(313,2),(326,1),(332,1),(342,3),(348,1), (353,2),(357,0),(363,1),(375,0),(381,4),(387,4),(396,5)}
T′₅₇={(8,4),(14,8),(20,4),(25,5),(37,0),(42,0),(50,13),(56,0),(60,4),(68,0), (83,0),(96,0),(102,6),(114,5),(121,10),(129,0),(139,10),(149,4),(156,10),(161,11), (172,2),(188,9),(197,0),(201,13),(210,9),(214,11),(222,1),(237,8),(240,1),(248,13), (258,5),(261,14),(270,9),(282,6),(291,5),(299,5),(303,5),(310,9),(324,5),(339,7), (345,13),(352,12),(357,8),(370,13),(380,4),(392,10)}
T′₅₈={(5,12),(10,12),(18,10),(23,13),(35,12),(41,8),(49,1),(55,9),(59,14), (67,0),(78,6),(91,8),(101,8),(110,2),(119,1),(127,15),(137,9),(148,13),(153,4), (160,1),(169,12),(187,6),(194,5),(200,7),(206,13),(212,3),(221,2),(236,7),(239,3), (245,13),(251,7),(260,7),(269,5),(279,5),(284,0),(296,11),(302,2),(306,5),(314,13), (333,6),(341,14),(349,0),(355,1),(364,13),(379,12),(383,5),(395,8)}
T′₅₉={(0,4),(9,8),(17,0),(21,9),(32,9),(40,12),(46,1),(52,12),(57,6),(61,10), (76,4),(89,6),(98,0),(105,1),(117,15),(123,5),(134,7),(141,1),(151,9),(158,11), (166,1),(179,11),(189,2),(198,1),(205,2),(211,1),(220,1),(234,1),(238,1),(242,7), (250,5),(259,3),(263,13),(274,0),(283,4),(295,3),(301,6),(305,4),(312,4),(332,12), (340,13),(348,12),(354,0),(361,0),(372,14),(381,11),(394,10)} (31)
T′₀={(5,12),(12,4),(20,9),(38,0),(56,9),(71,12),(82,13),(90,15),(99,15), (111,15),(121,8),(129,9),(152,13),(161,7),(168,11),(177,13),(194,8),(202,9),(213,1), (224,14),(232,5),(245,9),(260,5),(273,4),(284,8),(292,4),(309,5),(326,0),(337,1), (343,1),(351,4),(365,12),(375,12),(383,0),(391,3)}
T′₁={(3,10),(9,8),(19,0),(35,1),(48,13),(63,8),(78,2),(88,1),(97,0),(110,13), (120,9),(128,15),(150,12),(159,13),(165,13),(175,12),(193,1),(200,9),(212,13), (223,13),(228,7),(241,7),(255,5),(272,4),(278,5),(291,2),(302,4),(312,5),(335,4), (342,10),(350,1),(364,0),(373,4),(379,0),(389,0)}
T′₂={(2,8),(7,12),(15,0),(34,8),(46,0),(62,1),(77,3),(85,9),(93,12),(107,14), (117,7),(127,14),(134,12),(158,9),(164,4),(172,12),(185,9),(199,3),(207,13),(217,3), (227,5),(240,7),(251,5),(268,4),(276,6),(289,6),(299,4),(311,3),(334,9),(340,14), (349,9),(361,8),(369,2),(377,0),(388,4)}
T′₃={(1,8),(6,8),(14,12),(26,4),(45,13),(57,3),(75,8),(83,13),(92,4),(104,5), (116,9),(125,15),(131,5),(156,1),(163,5),(171,9),(180,5),(195,9),(204,3),(214,13), (226,10),(236,14),(250,12),(267,4),(275,9),(288,1),(297,12),(310,4),(330,10), (338,6),(347,4),(357,14),(367,6),(376,12),(384,1)}
T′₄={(10,8),(17,1),(30,12),(46,12),(65,13),(81,13),(100,9),(112,13),(128,5), (137,0),(155,0),(161,0),(174,8),(180,8),(188,8),(196,0),(202,1),(214,4),(226,12), (239,0),(251,5),(263,0),(270,5),(285,12),(295,5),(308,5),(315,13),(323,1),(338,1), (347,5),(363,5),(373,13),(382,1),(392,5),(397,5)}
T′₅={(8,9),(14,8),(29,9),(45,0),(60,0),(78,12),(99,5),(111,5),(127,12), (136,1),(143,4),(160,0),(170,12),(178,8),(187,8),(192,1),(200,0),(210,1),(224,0), (236,8),(249,1),(259,9),(267,1),(282,0),(292,8),(306,9),(312,13),(322,5),(328,5), (346,13),(361,1),(372,5),(380,4),(390,5),(395,9)}
T′₆={(5,0),(13,4),(24,13),(37,4),(56,4),(77,13),(94,5),(107,4),(125,0), (134,1),(141,12),(158,9),(169,9),(177,4),(186,12),(191,8),(199,8),(207,8),(221,9), (233,1),(246,1),(257,9),(266,0),(280,8),(291,0),(305,12),(311,1),(321,12),(327,13), (343,5),(356,13),(369,5),(376,1),(389,13),(394,8)}
T′₇={(2,4),(12,8),(23,12),(33,9),(47,8),(73,13),(89,12),(102,9),(117,4), (133,12),(138,12),(157,12),(163,8),(176,4),(182,8),(189,8),(198,8),(204,4),(217,2), (228,9),(241,0),(256,8),(264,1),(271,9),(286,1),(302,12),(309,4),(316,1),(324,5), (342,12),(349,0),(367,5),(374,4),(387,5),(393,1)}
T′₈={(5,3),(15,0),(22,11),(39,5),(46,15),(53,14),(61,11),(74,14),(88,5), (95,6),(105,6),(130,12),(142,11),(160,7),(182,12),(191,6),(205,8),(224,0),(233,0), (239,1),(248,1),(253,9),(264,7),(274,8),(287,1),(297,5),(304,11),(314,15),(322,10), (331,11),(345,0),(360,1),(375,9),(387,1),(399,13)}
T′₉={(3,3),(13,3),(21,6),(38,6),(44,3),(51,7),(59,6),(73,10),(86,14),(92,10), (104,6),(128,3),(139,6),(156,14),(176,12),(186,9),(204,0),(217,4),(230,4),(238,8), (247,12),(251,1),(262,3),(271,1),(286,11),(296,7),(302,11),(309,15),(317,10), (327,15),(335,9),(358,9),(371,3),(379,1),(396,15)}
T′₁₀={(2,14),(12,11),(20,13),(36,13),(43,0),(48,13),(58,8),(71,10),(85,7), (90,2),(103,4),(115,8),(136,2),(155,0),(171,10),(185,12),(199,10),(213,1),(228,0), (236,1),(246,13),(250,4),(258,5),(270,4),(280,2),(294,12),(301,7),(308,11),(316,14), (326,8),(334,5),(351,9),(363,3),(378,2),(394,13)}
T′₁₁={(1,2),(6,10),(19,13),(33,10),(40,12),(47,3),(57,15),(70,7),(79,1),(89,5), (99,14),(111,10),(133,4),(145,14),(168,5),(183,8),(193,15),(207,5),(226,0),(234,12), (240,3),(249,8),(254,0),(268,6),(277,5),(293,3),(299,7),(306,3),(315,8),(325,11), (333,1),(348,13),(362,15),(376,7),(389,11)}
T′₁₂={(7,13),(18,3),(32,4),(53,1),(64,15),(75,14),(90,8),(100,8),(105,15), (110,5),(121,12),(132,11),(140,9),(152,6),(164,12),(171,12),(181,6),(188,1), (199,15),(215,6),(235,2),(248,0),(256,0),(266,4),(281,2),(292,10),(299,15),(314,7), (327,0),(346,3),(353,1),(369,15),(374,6),(386,3),(399,10)}
T′₁₃={(4,7),(15,5),(30,7),(50,5),(60,8),(71,9),(85,10),(96,11),(104,4), (109,13),(116,0),(131,9),(137,11),(149,0),(162,0),(168,9),(180,5),(184,11),(197,6), (213,15),(234,6),(247,6),(255,2),(265,2),(280,4),(291,2),(297,0),(306,3),(325,10), (338,8),(352,8),(366,10),(373,13),(382,5),(396,8)}
T′₁₄={(3,15),(14,6),(29,12),(40,5),(55,8),(70,5),(81,5),(95,9),(103,0), (108,13),(113,9),(124,13),(136,12),(145,9),(159,5),(167,1),(176,0),(183,10),(190,9), (209,11),(232,0),(244,2),(254,2),(260,7),(278,0),(288,10),(295,4),(304,2),(322,11), (336,6),(351,9),(364,10),(371,15),(377,14),(389,7)}
T′₁₅={(2,4),(9,8),(22,4),(36,14),(54,11),(68,12),(77,2),(91,11),(101,5), (107,13),(111,9),(123,12),(133,1),(144,12),(155,7),(166,6),(174,5),(182,0),(189,1), (203,6),(223,3),(240,10),(253,6),(257,10),(274,0),(286,2),(294,1),(301,2),(321,1), (334,11),(348,3),(355,11),(370,14),(376,15),(387,2)}
T′₁₆={(16,9),(21,1),(31,1),(39,0),(55,0),(72,11),(78,3),(87,8),(99,7),(114,3), (120,15),(132,0),(147,2),(167,6),(175,10),(188,10),(200,3),(208,12),(217,0), (228,12),(242,14),(255,8),(261,7),(268,8),(278,15),(289,11),(298,4),(315,1),(324,5), (342,13),(356,4),(360,5),(365,0),(385,1),(398,11)}
T′₁₇={(5,9),(19,0),(29,8),(37,5),(50,9),(67,15),(76,5),(84,0),(96,2),(110,0), (119,10),(128,10),(143,2),(164,13),(174,8),(182,14),(199,9),(207,0),(213,7),(223,2), (241,6),(254,1),(259,11),(265,15),(277,11),(286,13),(297,3),(311,1),(320,4), (339,13),(348,7),(359,2),(364,8),(375,8),(393,5)}
T′₁₈={(1,5),(18,0),(28,9),(36,1),(44,13),(58,7),(74,11),(81,2),(93,7),(108,2), (118,10),(125,11),(140,3),(155,13),(173,11),(181,4),(195,10),(206,15),(212,7), (222,4),(231,12),(253,6),(257,6),(264,7),(275,15),(285,3),(295,0),(307,11),(318,5), (338,8),(347,13),(358,13),(363,1),(371,3),(389,0)}
T′₁₉={(0,0),(17,1),(26,0),(34,7),(40,15),(56,0),(73,1),(80,3),(89,5),(107,14), (117,1),(123,11),(138,10),(152,14),(171,2),(180,7),(191,2),(204,14),(211,6), (218,15),(230,9),(249,15),(256,1),(262,12),(269,12),(280,11),(293,13),(300,12), (316,6),(326,9),(346,5),(357,2),(361,0),(368,13),(388,3)}
T′₂₀={(12,3),(24,15),(35,10),(41,9),(49,7),(65,0),(84,5),(105,14),(114,14), (121,8),(131,14),(137,10),(144,0),(159,12),(176,0),(193,13),(197,4),(214,5),(222,3), (228,9),(236,3),(247,4),(263,8),(274,10),(282,8),(294,10),(308,2),(319,0),(332,4), (338,11),(354,5),(360,1),(368,14),(379,6),(398,15)}
T′₂₁={(8,11),(21,9),(34,11),(40,7),(45,14),(61,13),(83,15),(101,5),(113,6), (119,0),(129,1),(136,8),(142,15),(154,11),(175,3),(186,3),(196,3),(210,0),(219,4), (227,9),(235,5),(245,0),(262,8),(272,1),(281,0),(291,14),(303,10),(317,0),(329,2), (336,0),(353,3),(359,6),(366,2),(378,12),(388,14)}
T′₂₂={(6,9),(19,14),(31,14),(38,6),(44,4),(59,5),(76,10),(88,12),(109,6), (117,3),(124,4),(135,9),(141,8),(151,7),(169,15),(184,1),(195,4),(209,15),(218,14), (225,5),(232,5),(243,8),(250,0),(267,1),(278,15),(288,14),(300,11),(313,8),(327,1), (335,2),(351,4),(356,10),(363,2),(376,10),(385,7)}
T′₂₃={(1,14),(18,10),(25,10),(37,8),(43,3),(53,7),(72,11),(86,11),(108,5), (115,13),(122,7),(132,1),(139,3),(148,12),(166,1),(183,13),(194,3),(205,4),(217,10), (223,0),(229,9),(241,1),(248,5),(266,5),(275,1),(287,15),(298,1),(309,10),(325,2), (333,2),(350,10),(355,5),(362,15),(373,6),(383,1)}
T′₂₄={(11,15),(22,5),(33,1),(42,8),(55,11),(71,9),(78,10),(86,10),(95,11), (104,9),(112,9),(120,1),(134,4),(147,0),(155,10),(171,3),(188,0),(201,10),(208,13), (217,8),(231,5),(241,6),(261,3),(266,1),(279,2),(290,6),(296,4),(313,10),(326,0), (342,7),(358,11),(371,3),(388,2),(395,2),(399,3)}
T′₂₅={(7,10),(16,11),(31,12),(38,13),(54,14),(70,1),(76,8),(84,9),(93,0), (103,1),(111,1),(119,1),(131,1),(139,15),(153,2),(170,11),(185,3),(198,1),(207,14), (216,7),(229,5),(237,6),(258,6),(264,2),(273,14),(288,0),(295,10),(308,12),(325,1), (341,10),(356,1),(370,3),(387,4),(392,5),(398,7)}
T₂₆={(6,3),(15,3),(28,12),(36,13),(44,1),(69,9),(75,9),(81,1),(88,13),(102,1), (110,4),(117,2),(130,1),(138,9),(150,11),(168,7),(180,11),(196,3),(205,7),(215,2), (227,14),(234,0),(248,0),(263,5),(269,6),(287,15),(294,1),(302,11),(321,14),(336,4), (355,13),(369,9),(385,2),(391,8),(397,5)}
T′₂₇={(4,8),(13,11),(27,3),(34,13),(43,12),(57,12),(73,14),(80,8),(87,13), (101,0),(107,1),(114,9),(122,8),(135,10),(149,3),(165,13),(173,8),(193,10),(203,11), (211,4),(220,3),(233,14),(242,6),(262,14),(268,12),(283,14),(292,2),(300,4),(317,6), (332,10),(343,7),(365,7),(377,2),(390,7),(396,1)}
T′₂₈={(4,8),(18,14),(27,14),(38,14),(52,14),(63,14),(74,6),(84,0),(107,13), (121,2),(131,7),(139,1),(147,5),(154,10),(173,11),(184,3),(189,6),(204,3),(215,6), (225,1),(232,13),(242,10),(248,9),(255,9),(263,0),(270,9),(285,9),(292,5),(308,1), (318,12),(328,9),(341,0),(360,10),(375,3),(396,11)}
T′₂₉={(3,6),(17,10),(25,14),(36,3),(48,0),(62,12),(69,0),(82,4),(103,2), (118,12),(129,6),(138,6),(146,3),(153,9),(168,0),(183,8),(187,2),(203,11),(209,1), (224,0),(231,7),(238,2),(247,1),(253,0),(262,9),(269,0),(281,9),(290,8),(306,0), (317,3),(324,12),(340,7),(348,11),(372,11),(390,6)}
T′₃₀={(1,1),(15,15),(22,8),(33,10),(45,11),(59,3),(66,14),(80,2),(99,8), (110,1),(125,3),(137,10),(145,7),(151,2),(165,3),(179,9),(186,2),(196,4),(206,5), (222,1),(229,3),(237,2),(245,5),(251,3),(259,1),(268,1),(276,9),(289,11),(297,1), (314,0),(322,12),(339,13),(347,15),(368,5),(382,4)}
T′₃₁={(0,15),(10,9),(20,9),(31,4),(41,10),(55,10),(64,3),(77,5),(87,6), (108,10),(122,7),(132,7),(142,15),(148,11),(157,0),(175,15),(185,1),(194,9), (205,11),(219,1),(226,1),(234,0),(243,11),(250,2),(256,4),(267,3),(271,8),(286,10), (293,10),(309,1),(320,13),(336,12),(346,15),(367,0),(378,6)}
T′₃₂={(11,9),(26,4),(34,2),(51,8),(57,2),(64,2),(69,5),(83,12),(98,4),(103,8), (118,4),(130,2),(145,0),(152,14),(162,2),(170,10),(181,2),(196,14),(205,0),(214,3), (239,10),(244,14),(254,8),(277,8),(288,7),(305,14),(312,6),(323,11),(330,11), (345,5),(352,7),(361,14),(372,9),(382,6),(394,7)}
T′₃₃={(8,11),(25,1),(33,5),(50,3),(56,7),(63,12),(68,12),(80,9),(97,3),(101,2), (109,5),(127,5),(140,4),(151,6),(161,4),(169,1),(179,0),(193,3),(203,0),(211,2), (233,9),(243,8),(251,0),(276,10),(285,3),(304,12),(310,7),(318,13),(329,10), (344,14),(350,13),(357,3),(370,12),(380,12),(392,0)}
T′₃₄={(3,13),(24,3),(31,4),(44,0),(54,8),(60,1),(67,1),(72,7),(95,13),(100,8), (108,8),(126,11),(138,9),(149,6),(158,14),(166,1),(178,8),(190,0),(202,1),(209,2), (231,9),(242,6),(247,9),(268,7),(284,10),(295,0),(308,10),(317,14),(328,6),(340,13), (347,6),(355,4),(367,3),(377,7),(389,9)}
T′₃₅={(2,8),(16,14),(28,12),(39,4),(52,14),(58,8),(66,3),(70,7),(90,9),(99,12), (105,2),(119,5),(133,0),(147,1),(156,1),(163,1),(177,2),(186,4),(201,2),(207,11), (227,3),(241,2),(245,13),(257,0),(282,10),(292,4),(307,13),(314,12),(324,11), (335,11),(346,7),(354,3),(365,5),(376,4),(386,2)}
T′₃₆={(5,15),(21,2),(33,10),(54,8),(70,0),(80,10),(86,6),(96,10),(111,1), (123,10),(141,7),(152,4),(159,13),(171,4),(177,9),(190,9),(199,0),(204,14),(220,8), (233,6),(247,1),(257,7),(265,11),(271,11),(286,2),(302,2),(316,8),(331,13),(339,9), (346,6),(363,4),(369,2),(378,11),(390,10),(397,11)}
T′₃₇={(4,1),(20,2),(32,6),(53,7),(68,0),(74,3),(84,13),(92,5),(102,13),(118,4), (139,2),(149,2),(158,1),(170,6),(175,1),(184,12),(198,12),(202,12),(216,15), (230,12),(246,15),(255,12),(264,14),(270,8),(279,6),(298,14),(313,15),(330,10), (335,6),(345,10),(362,14),(368,1),(377,3),(387,4),(395,0)}
T′₃₈={(3,11),(10,0),(26,1),(46,2),(62,5),(73,9),(82,10),(91,4),(99,2),(116,5), (138,8),(146,14),(157,3),(167,0),(174,1),(181,4),(196,0),(201,0),(208,5),(229,15), (237,8),(252,10),(263,2),(269,0),(275,10),(296,10),(311,3),(320,2),(333,2),(344,6), (351,10),(367,4),(375,1),(384,11),(394,1)}
T′₃₉={(1,2),(9,7),(25,8),(37,4),(58,3),(72,0),(81,2),(89,0),(97,5),(115,5), (132,6),(143,7),(154,4),(162,0),(172,4),(178,14),(195,8),(200,0),(207,3),(226,0), (235,8),(248,7),(258,3),(267,2),(272,8),(293,0),(307,7),(319,14),(332,8),(340,2), (350,10),(366,14),(373,0),(380,0),(392,3)}
T′₄₀={(6,11),(11,4),(18,2),(38,2),(49,0),(53,6),(62,6),(79,2),(92,8),(101,14), (112,2),(124,14),(133,10),(145,8),(157,0),(165,8),(174,2),(192,14),(209,2),(222,2), (237,2),(252,14),(258,14),(273,14),(292,2),(303,14),(320,6),(330,4),(336,8),(343,4), (358,8),(376,4),(381,10),(391,8),(399,4)}
T′₄₁={(4,6),(10,0),(17,0),(34,8),(45,2),(52,6),(61,8),(74,2),(91,6),(98,10), (111,10),(122,2),(130,0),(144,8),(156,10),(161,0),(173,4),(191,14),(206,15),(220,6), (232,8),(245,0),(257,10),(267,2),(290,10),(301,2),(319,6),(326,6),(333,8),(340,0), (352,8),(372,0),(379,4),(389,8),(397,12)}
T′₄₂={(3,2),(9,10),(16,14),(31,6),(43,0),(51,12),(60,2),(69,8),(88,0),(97,0), (108,10),(116,10),(129,8),(143,0),(151,10),(159,10),(171,8),(184,2),(204,0),(219,6), (228,2),(244,12),(255,8),(262,8),(279,8),(299,10),(315,0),(324,2),(332,0),(339,0), (349,0),(367,14),(378,12),(388,0),(394,0)}
T′₄₃={(0,10),(8,0),(13,12),(30,10),(42,7),(50,8),(58,10),(67,0),(87,4),(93,2), (104,2),(115,2),(126,8),(135,2),(150,8),(158,10),(170,2),(182,14),(198,4),(216,12), (226,6),(240,14),(253,10),(259,4),(278,6),(296,6),(310,6),(323,8),(331,4),(337,10), (344,4),(366,8),(377,14),(383,12),(393,6)}
T′₄₄={(13,10),(20,10),(37,2),(48,12),(57,11),(73,3),(83,2),(93,11),(98,5), (113,4),(126,7),(138,1),(147,11),(151,3),(164,9),(169,7),(181,5),(193,3),(203,13), (215,1),(220,3),(227,6),(233,2),(244,6),(256,1),(276,8),(286,1),(303,2),(317,0), (327,0),(348,8),(360,8),(373,0),(390,10),(398,10)}
T′₄₅={(10,10),(18,10),(32,10),(45,0),(52,6),(66,0),(82,3),(92,11),(97,3), (109,3),(119,1),(135,5),(146,9),(150,3),(159,1),(168,3),(180,1),(192,3),(202,3), (213,3),(219,5),(226,3),(231,2),(239,1),(250,9),(275,2),(283,8),(300,2),(312,0), (323,0),(345,8),(359,12),(371,4),(388,1),(397,0)}
T′₄₆={(6,8),(16,2),(30,0),(42,14),(51,7),(61,1),(78,11),(90,10),(96,15), (108,0),(116,1),(131,3),(144,5),(149,1),(154,3),(167,3),(178,3),(186,5),(198,5), (212,9),(218,15),(225,1),(230,8),(238,2),(249,11),(265,2),(280,0),(296,2),(311,0), (319,0),(336,0),(358,8),(367,10),(375,8),(394,2)}
T₄₇={(4,2),(15,10),(23,6),(39,14),(49,2),(59,11),(75,14),(87,13),(95,1), (106,7),(115,7),(130,14),(142,7),(148,0),(153,3),(166,11),(170,9),(183,1),(197,3), (210,9),(217,1),(221,1),(229,12),(235,10),(246,5),(262,7),(279,14),(291,0),(309,0), (318,12),(329,0),(354,8),(364,0),(374,0),(391,10)}
T′₄₈={(11,13),(23,0),(30,11),(48,7),(63,3),(70,2),(83,7),(94,11),(101,1), (113,9),(121,1),(128,1),(142,5),(163,9),(176,1),(180,2),(189,0),(202,8),(219,2), (227,6),(241,0),(250,4),(256,0),(265,0),(277,2),(290,2),(311,2),(321,10),(325,0), (335,10),(344,10),(353,8),(360,1),(384,11),(397,3)}
T′₄₉={(8,1),(22,11),(28,9),(46,3),(62,3),(66,7),(82,5),(86,5),(100,15),(112,9), (120,1),(127,11),(141,9),(162,3),(172,5),(179,6),(187,7),(198,6),(218,0),(225,0), (240,2),(249,4),(254,8),(263,4),(273,4),(287,2),(307,10),(316,10),(324,0),(334,10), (341,14),(350,4),(359,12),(383,14),(393,13)}
T′₅₀={(7,3),(16,1),(27,8),(45,7),(59,1),(65,7),(80,3),(85,11),(98,9),(109,0), (117,5),(124,3),(140,9),(160,5),(167,11),(178,1),(185,0),(192,15),(208,10),(224,3), (239,3),(243,6),(252,8),(260,1),(272,0),(284,4),(301,10),(313,12),(323,12),(331,2), (337,10),(349,6),(358,2),(380,4),(392,2)}
T′₅₁={(1,3),(14,2),(24,15),(44,9),(50,3),(64,3),(76,1),(84,9),(97,5),(102,3), (114,7),(123,1),(134,3),(152,1),(165,1),(177,0),(181,1),(190,2),(206,2),(221,4), (233,6),(242,0),(251,8),(259,1),(270,1),(278,2),(293,6),(312,6),(322,12),(330,8), (336,8),(347,10),(355,0),(369,2),(391,3)}
T′₅₂={(11,11),(23,3),(31,9),(41,11),(48,7),(53,3),(71,1),(88,7),(102,7), (114,2),(123,3),(130,4),(146,3),(154,0),(162,6),(175,8),(190,9),(199,0),(209,0), (214,0),(222,0),(230,0),(246,0),(271,0),(282,0),(290,0),(304,8),(314,4),(319,12), (334,1),(350,1),(354,11),(368,9),(382,4),(399,4)}
T′₅₃={(9,1),(15,1),(27,1),(36,9),(47,1),(52,1),(69,13),(86,9),(97,2),(113,10), (122,1),(129,1),(144,1),(152,3),(160,10),(173,8),(183,6),(195,7),(208,6),(213,8), (221,10),(229,10),(243,9),(266,4),(281,6),(289,12),(300,0),(312,0),(318,10),(329,0), (347,9),(353,0),(365,11),(380,3),(386,1)}
T′₅₄={(7,0),(13,1),(25,3),(35,9),(43,3),(51,13),(65,15),(85,9),(96,3),(106,7), (120,2),(127,1),(142,11),(148,15),(157,9),(166,14),(179,11),(193,4),(206,1),(212,2), (218,1),(224,10),(236,2),(261,10),(276,10),(284,8),(298,10),(307,4),(316,2), (325,12),(342,4),(352,2),(364,13),(372,0),(385,1)}
T′₅₅={(5,3),(12,3),(24,3),(33,9),(42,3),(49,11),(58,5),(81,3),(94,11),(104,15), (115,3),(125,5),(140,3),(147,5),(156,2),(165,12),(176,2),(192,10),(205,2),(210,10), (216,10),(223,8),(231,2),(249,13),(273,6),(283,0),(293,2),(305,8),(315,0),(320,8), (338,3),(351,2),(361,9),(371,3),(384,0)}
T′₅₆={(19,2),(29,0),(40,8),(51,2),(58,2),(66,3),(77,0),(82,12),(92,11), (105,10),(112,15),(124,13),(133,11),(143,9),(155,1),(172,1),(184,8),(197,11), (211,3),(225,11),(240,3),(264,3),(276,5),(287,1),(298,2),(313,0),(319,12),(327,1), (334,2),(359,1),(370,10),(379,1),(386,4),(393,0),(399,0)}
T′₅₇={(12,0),(26,10),(34,0),(49,0),(55,0),(64,0),(75,9),(81,9),(91,15), (100,11),(110,2),(120,3),(132,3),(141,2),(153,7),(167,8),(183,0),(194,3),(208,13), (223,3),(238,3),(259,1),(275,9),(282,9),(294,0),(310,1),(318,0),(322,1),(331,0), (340,2),(368,8),(374,2),(384,4),(391,3),(398,10)}
T′₅₈={(6,0),(25,8),(32,0),(47,0),(54,8),(63,0),(72,8),(79,5),(90,1),(95,0), (109,11),(118,9),(126,15),(140,11),(146,11),(163,7),(178,1),(187,11),(206,3), (219,1),(237,9),(252,9),(274,9),(280,9),(293,9),(301,1),(316,12),(321,10),(330,0), (339,0),(365,5),(372,2),(382,8),(388,4),(396,2)}
T′₅₉={(0,8),(20,0),(30,2),(41,0),(52,2),(60,0),(69,1),(78,0),(87,4),(94,1), (106,10),(115,1),(125,0),(135,3),(145,9),(161,11),(175,15),(186,1),(201,3),(216,3), (235,9),(246,11),(273,2),(277,11),(289,5),(300,3),(315,10),(320,8),(328,2),(337,4), (362,8),(371,1),(381,8),(387,0),(395,2)}
T′₆₀={(11,0),(26,2),(32,3),(42,1),(59,0),(67,11),(74,10),(87,1),(106,9), (121,11),(131,9),(136,1),(149,1),(158,3),(165,9),(174,3),(185,1),(195,9),(208,1), (215,11),(227,11),(243,9),(260,0),(272,1),(283,9),(291,11),(310,2),(323,3),(337,8), (342,0),(349,10),(359,0),(374,10),(383,8),(395,0)}
T′₆₁={(8,8),(21,8),(29,0),(39,11),(56,0),(65,3),(72,3),(84,8),(102,1),(118,0), (127,1),(135,1),(142,1),(157,1),(163,1),(173,1),(184,1),(194,9),(203,3),(214,3), (221,1),(234,3),(253,1),(271,11),(282,2),(289,8),(306,10),(319,9),(333,2),(341,10), (345,0),(356,0),(370,0),(381,8),(391,8)}
T′₆₂={(4,10),(16,0),(28,2),(38,2),(54,10),(63,1),(71,3),(82,0),(94,8),(116,8), (126,9),(134,1),(140,1),(156,1),(162,11),(172,0),(181,1),(191,3),(201,1),(212,8), (219,8),(232,3),(252,2),(265,9),(281,10),(288,10),(303,3),(318,11),(330,1),(339,2), (344,0),(353,8),(366,8),(379,10),(390,2)}
T′₆₃={(2,2),(13,0),(27,10),(36,0),(51,2),(61,0),(68,0),(79,1),(88,3),(112,11), (124,3),(132,0),(139,1),(150,1),(159,1),(166,2),(179,1),(189,1),(197,0),(209,8), (218,11),(230,3),(244,11),(261,11),(273,1),(285,8),(294,2),(317,2),(329,1),(338,2), (343,2),(350,0),(362,3),(377,0),(384,0)}
T′₆₄={(19,6),(30,7),(54,4),(64,6),(72,0),(77,4),(89,6),(96,5),(113,14), (126,12),(137,14),(148,2),(155,2),(164,2),(182,12),(190,1),(210,12),(218,3), (228,10),(237,1),(252,0),(261,1),(270,1),(279,0),(288,1),(298,1),(302,1),(308,1), (321,5),(331,10),(337,5),(356,5),(366,5),(385,1),(398,6)}
T′₆₅={(17,3),(29,6),(47,14),(63,1),(71,13),(76,10),(85,7),(94,3),(106,2), (122,2),(136,10),(147,4),(153,14),(163,7),(176,1),(189,0),(203,5),(216,0),(225,7), (236,0),(245,1),(260,7),(269,5),(277,5),(285,9),(297,9),(301,0),(305,1),(315,1), (328,5),(335,7),(355,4),(362,6),(381,1),(397,5)}
T′₆₆={(11,5),(27,6),(42,6),(56,2),(67,2),(75,12),(80,0),(93,6),(100,0),(118,2), (133,6),(144,14),(151,0),(162,2),(169,3),(187,3),(197,1),(212,0),(224,4),(235,8), (244,7),(258,3),(266,1),(276,2),(284,7),(295,7),(300,3),(304,3),(313,1),(326,5), (334,1),(344,4),(358,4),(374,4),(395,4)}
T′₆₇={(7,4),(22,5),(35,14),(55,4),(66,0),(73,0),(79,0),(90,4),(98,4),(116,4), (128,6),(141,2),(150,0),(157,2),(167,1),(185,4),(191,11),(211,0),(220,11),(232,3), (238,9),(256,1),(265,7),(275,3),(280,7),(290,1),(299,3),(303,1),(311,0),(325,2), (332,13),(341,5),(357,5),(373,4),(386,13)}
T′₆₈={(12,1),(22,5),(28,9),(39,1),(44,10),(57,1),(68,5),(85,1),(94,4),(105,4), (112,7),(126,1),(145,12),(160,4),(172,0),(188,0),(195,4),(213,11),(231,0),(244,4), (252,4),(264,0),(272,0),(284,0),(299,10),(307,7),(322,7),(331,10),(346,0),(354,2), (363,3),(369,5),(382,7),(386,0),(399,1)}
T′₆₉={(7,2),(21,5),(27,2),(37,1),(43,1),(53,5),(65,5),(83,2),(92,3),(104,0), (109,7),(120,7),(144,4),(154,5),(169,2),(179,7),(194,8),(201,5),(230,4),(243,0), (249,4),(258,0),(271,0),(283,0),(298,7),(306,4),(321,1),(329,2),(345,3),(353,5), (362,0),(368,5),(381,2),(385,6),(396,1)}
T′₇₀={(2,5),(19,1),(24,3),(35,13),(41,0),(47,0),(64,5),(79,1),(91,1),(103,3), (107,5),(119,5),(143,2),(153,5),(168,0),(178,4),(192,8),(200,0),(229,1),(242,4), (247,4),(254,4),(269,7),(281,2),(289,2),(305,3),(320,14),(328,6),(343,0),(352,11), (359,1),(365,6),(378,5),(384,12),(393,3)}
T′₇₁={(0,1),(14,5),(23,7),(32,1),(40,1),(46,1),(62,11),(77,7),(86,4),(95,4), (106,8),(113,2),(135,1),(146,13),(164,6),(177,13),(190,4),(196,0),(215,4),(235,0), (246,6),(253,4),(266,4),(274,4),(287,6),(304,1),(309,6),(327,7),(333,6),(351,1), (357,2),(364,7),(374,0),(383,3),(390,3)}
T′₇₂={(14,2),(26,2),(39,4),(48,3),(61,2),(67,14),(74,4),(89,6),(100,4),(122,4), (128,0),(136,4),(148,4),(161,4),(174,1),(188,6),(202,0),(216,1),(223,3),(239,5), (260,13),(272,5),(279,1),(287,0),(297,1),(307,5),(323,3),(332,1),(343,3),(352,1), (357,0),(366,0),(380,1),(387,4),(398,5)}
T′₇₃={(10,6),(24,7),(35,5),(47,7),(60,2),(66,5),(70,4),(79,2),(98,7),(114,0), (125,14),(134,6),(146,5),(160,4),(173,0),(187,1),(200,7),(215,1),(222,5),(238,0), (257,1),(268,1),(278,5),(285,3),(296,1),(305,5),(314,1),(329,0),(342,3),(349,1), (356,9),(363,1),(378,5),(386,5),(396,5)}
T′₇₄={(9,3),(23,7),(29,12),(43,3),(50,0),(65,0),(69,4),(76,2),(93,6),(106,1), (124,14),(130,4),(143,5),(154,0),(170,4),(182,5),(192,1),(211,9),(221,1),(234,5), (255,5),(267,7),(277,3),(283,6),(294,1),(304,0),(313,2),(328,2),(341,0),(348,1), (354,1),(361,13),(375,0),(385,0),(393,7)}
T′₇₅={(0,2),(17,6),(28,5),(41,5),(49,4),(62,4),(68,2),(75,6),(91,4),(103,4), (123,6),(129,4),(137,4),(150,5),(164,0),(177,4),(191,0),(210,3),(220,0),(225,0), (254,0),(261,7),(274,3),(281,1),(290,1),(303,1),(310,0),(326,1),(337,0),(344,1), (353,2),(360,1),(370,5),(381,4),(392,6)}
T′₇₆={(9,8),(18,10),(25,5),(40,12),(49,1),(56,0),(61,10),(78,6),(96,0),(105,1), (119,1),(129,0),(141,1),(153,4),(161,11),(179,11),(194,5),(201,13),(211,1),(221,2), (237,8),(242,7),(251,7),(261,14),(274,0),(284,0),(299,5),(305,4),(314,13),(339,7), (348,12),(355,1),(370,13),(381,11),(395,8)}
T′₇₇={(8,4),(17,0),(23,13),(37,0),(46,1),(55,9),(60,4),(76,4),(91,8),(102,6), (117,15),(127,15),(139,10),(151,9),(160,1),(172,2),(189,2),(200,7),(210,9),(220,1), (236,7),(240,1),(250,5),(260,7),(270,9),(283,4),(296,11),(303,5),(312,4),(333,6), (345,13),(354,0),(364,13),(380,4),(394,10)}
T′₇₈={(5,12),(14,8),(21,9),(35,12),(42,0),(52,12),(59,14),(68,0),(89,6), (101,8),(114,5),(123,5),(137,9),(149,4),(158,11),(169,12),(188,9),(198,1),(206,13), (214,11),(234,1),(239,3),(248,13),(259,3),(269,5),(282,6),(295,3),(302,2),(310,9), (332,12),(341,14),(352,12),(361,0),(379,12),(392,10)}
T′₇₉={(0,4),(10,12),(20,4),(32,9),(41,8),(50,13),(57,6),(67,0),(83,0),(98,0), (110,2),(121,10),(134,7),(148,13),(156,10),(166,1),(187,6),(197,0),(205,2),(212,3), (222,1),(238,1),(245,13),(258,5),(263,13),(279,5),(291,5),(301,6),(306,5),(324,5), (340,13),(349,0),(357,8),(372,14),(383,5)} (32)

FIG. 5 is a block diagram illustrating a quasi-cyclic LDPC encoder according to an embodiment of the present invention.

Referring to FIG. 5, a quasi-cyclic LDPC encoder 111 includes an information word vector convertor 511, a parity vector generator 513, and a quasi-cyclic LDPC codeword vector generator 515.

Upon inputting an information word vector, the information word vector convertor 511 generates a converted information word vector by performing a conversion operation on the information word vector, based on control information and a parity check matrix, and outputs the converted information word vector to the parity vector generator 513. For example, the conversion operation may be a padding operation in which zeros are inserted into the information word vector, based on the control information and the parity check matrix.

The parity vector generator 513 converts the information word vector output from the information word vector convertor 511 into a parity vector based on the control information and the parity check matrix, and outputs the parity vector to the quasi-cyclic LDPC codeword vector generator 515. The quasi-cyclic LDPC codeword vector generator 515 generates a quasi-cyclic LDPC codeword vector by concatenating the parity vector and the information word vector, based on the control information.

Alternatively, the quasi-cyclic LDPC codeword vector generator 515 may generate the quasi-cyclic LDPC codeword vector by concatenating a converted parity vector and the information word vector. The converted parity vector is generated by puncturing the parity vector.

FIG. 6 is a block diagram illustrating a quasi-cyclic LDPC code decoder included in a signal reception apparatus in an MMT system according to an embodiment of the present invention.

Referring to FIG. 6, a quasi-cyclic LDPC code decoder includes a quasi-cyclic LDPC decoder 611 and a parity check matrix generator 613.

A received vector is input to the quasi-cyclic LDPC decoder 611. For example, the received vector may be a quasi-cyclic LDPC vector output from a quasi-cyclic LDPC encoder. Control information including (k,n,m) information is input to the quasi-cyclic LDPC decoder 111. The control information is described before, so the detailed description will be omitted herein.

The parity check matrix generator 613 also inputs the control information, converts a prestored base matrix into a parity check matrix, based on the control information, and outputs the parity check matrix to the quasi-cyclic LDPC decoder 611.

The quasi-cyclic LDPC decoder 611 generates a recovered information vector by quasi-cyclic LDPC decoding the received vector, based on the control information.

Although FIG. 6 illustrates the parity check matrix generator 613 generating the parity check matrix and outputting the parity check matrix to the quasi-cyclic LDPC decoder 611, alternatively, the quasi-cyclic LDPC decoder 611 may prestore the parity check matrix, and in this case, the parity check matrix generator 613 is not utilized.

Although FIG. 6 illustrates the control information being input from the outside to the quasi-cyclic LDPC decoder 611 and the parity check matrix generator 613, alternatively, the quasi-cyclic LDPC decoder 611 and the parity check matrix generator 613 may prestore the control information.

While the quasi-cyclic LDPC decoder 611 and the parity check matrix generator 613 are illustrated in FIG. 6 as separate units, alternatively, these components may be incorporated into a single unit.

Although not illustrated, the signal reception apparatus includes the quasi-cyclic LDPC code decoder and a receiver, and the quasi-cyclic LDPC code decoder and the receiver may be incorporated into a single unit.

FIG. 7 is a flow chart illustrating an operation process of a quasi-cyclic LDPC code generator included in a signal transmission apparatus in an MMT system according to an embodiment of the present invention.

Referring to FIG. 7, in step 711, a quasi-cyclic LDPC code generator receives an information word vector and control information. In step 713, the quasi-cyclic LDPC code generator generates a quasi-cyclic LDPC codeword vector by performing a quasi-cyclic LDPC encoding operation on the information word vector, based on the control information. The quasi-cyclic LDPC codeword vector includes an information word vector including k information word symbols and a parity vector including m parity symbols.

FIG. 8 is a flow chart illustrating an operation process of a quasi-cyclic LDPC code decoder included in a signal reception apparatus in an MMT system according to an embodiment of the present invention.

Referring to FIG. 8, in step 811, a quasi-cyclic LDPC code decoder receives a received vector and control information. In step 813, the quasi-cyclic LDPC code decoder recovers an information word vector by quasi-cyclic LDPC decoding the received vector using the control information and a parity check matrix.

As is apparent from the foregoing description, various embodiments of the present invention provide quasi-cyclic LDPC code transmission and reception supporting various codeword lengths and code rates in a multimedia communication system. Further, the above-described embodiments of the present invention provide quasi-cyclic LDPC code encoding and decoding that support various codeword lengths using a scaling scheme and a shortening scheme in a multimedia communication system, and support various code rates using a row separation scheme or a row merge scheme, and a puncturing scheme in a multimedia communication system.

While the present invention has been shown and described with reference to certain embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the present invention as defined by the appended claims and their equivalents.

Claims

1. A method for transmitting a quasi-cyclic Low Density Parity Check (LDPC) code by a signal transmission apparatus in a multimedia communication system, comprising:

generating a quasi-cyclic LDPC code; and

transmitting the quasi-cyclic LDPC code to a signal reception apparatus,

wherein the quasi-cyclic LDPC code is generated by encoding an information word vector using a child parity check matrix which is generated by performing one of a scaling operation, a row separation operation and a row merge operation on a parent parity check matrix, and

wherein the scaling operation is an operation in which a size of the child parity check matrix is determined, the row separation operation is an operation in which each of rows included in the parent parity check matrix is separated, and the row merge operation is an operation in which the rows included in the parent parity check matrix are merged.

2. The method as claimed in claim 1, wherein the parent parity check matrix includes an information part mapping to the information word vector, and is generated using a base matrix.

3. The method as claimed in claim 1, wherein the row separation operation is an operation in which each of the rows included in the parent parity check matrix is separated based on a preset row separation criterion, the parent parity check matrix is generated using a base matrix, and the row separation criterion is determined using information related to permutation matrixes included in the parent parity check matrix and information related to a specific permutation matrix among permutation matrixes included in the base matrix.

4. The method as claimed in claim 3, wherein the information related to the permutation matrixes included in the parent parity check matrix includes information on a sequence indicating locations of which the permutation matrixes included in the parent parity check matrix are arranged on the parent parity check matrix, and the information related to the specific permutation matrix among permutation matrixes included in the base matrix includes information on an exponent of a permutation matrix arranged in a location at which a specific row among rows included in the base matrix and a specific column among columns included in the base matrix cross.

5. The method as claimed in claim 1, wherein the row separation operation is an operation in which each of the rows included in the parent parity check matrix is separated based on a preset row separation criterion, and the row separation criterion is expressed as:

1) T0=T′0∪T′1∪... ∪T′n−1

2) ∩T′j={ } for all 0≦≠j<n

where Ti denotes a sequence indicating locations of which permutation matrixes are arranged on the parent parity check matrix, and T′i denotes a sequence indicating a location on which a permutation matrix is arranged on a related row among n rows if the Ti is separated into the n rows.

6. The method as claimed in claim 1, wherein the row separation operation is an operation in which each of the rows included in the parent parity check matrix is separated based on a preset row separation criterion, if the row separation criterion indicates each of all rows included in the parent parity check matrix is separated into n rows, a jth row is generated by separating Ti={(ti,0,ei,0),(ti,1,ei,1),..., (Di−1,Di−1)} is mapped to an (nxi+j)th row included in a parity check matrix generated by performing the row separation operation on the parent parity check matrix, and the (nxi+j)th row included in a parity check matrix is expressed as:

x+j={()|0≦k<Di,k mod n=(n−1−j)},

where Ti denotes a sequence indicating locations of which permutation matrixes are arranged on the parent parity check matrix, T′i denotes a sequence indicating location on which a permutation matrix is arranged on a related row among n rows if the Ti is separated into the n rows, and ti,Di−1 denotes a location of a permutation matrix included in an ith row among the permutation matrixes included in the parent parity check matrix, and ei,Di−1 denotes an exponent of a permutation matrix arrayed in a location indicated by ti,Di−1.

7. The method as claimed in claim 6, wherein the parent parity check matrix is generated using a base matrix,

wherein if a number of rows included in the base matrix is 2 and a number of columns included in the base matrix is 10, the parent parity check matrix is expressed as: T0={(0,0),(1,7),(2,4),(3,1),(4,6)} T1={(5,3),(6,0),(7,1),(8,6),(9,7)},

an 8×8 permutation matrix is substituted for each element included in the base matrix, the base matrix is expressed as: R0={0,1,2,3,4} R1={5,6,7,8,9},

where Ri denotes a sequence indicating locations at which elements have a ‘1’ value on the base matrix, ri,Di−1 as elements included in the Ri denotes an element included in the Ri and an index of a column with a value 1 on an ith row, and the Di denotes a degree of the ith row.

8. A method for receiving a quasi-cyclic Low Density Parity Check (LDPC) code by a signal reception apparatus in a multimedia communication system, comprising:

receiving a quasi-cyclic LDPC code; and

recovering an information word vector by decoding the quasi-cyclic LDPC code,

wherein the quasi-cyclic LDPC code is generated by encoding an information word vector using a child parity check matrix which is generated by performing one of a scaling operation, a row separation operation and a row merge operation on a parent parity check matrix, and

wherein the scaling operation is an operation in which a size of the child parity check matrix is determined, the row separation operation is an operation in which each of rows included in the parent parity check matrix is separated, and the row merge operation is an operation in which the rows included in the parent parity check matrix are merged.

9. The method as claimed in claim 8, wherein the parent parity check matrix includes an information part mapping to the information word vector and is generated using a base matrix.

10. The method as claimed in claim 8, wherein the row separation operation is an operation in which each of the rows included in the parent parity check matrix is separated based on a preset row separation criterion, the parent parity check matrix is generated using a base matrix, and the row separation criterion is determined using information related to permutation matrixes included in the parent parity check matrix and information related to a specific permutation matrix among permutation matrixes included in the base matrix.

11. The method as claimed in claim 10, wherein the information related to the permutation matrixes included in the parent parity check matrix includes information on a sequence indicating locations of which the permutation matrixes included in the parent parity check matrix are arranged on the parent parity check matrix, and the information related to the specific permutation matrix among permutation matrixes included in the base matrix includes information on an exponent of a permutation matrix arranged in a location at which a specific row among rows included in the base matrix and a specific column among columns included in the base matrix cross.

12. The method as claimed in claim 8, wherein the row separation operation is an operation in which each of the rows included in the parent parity check matrix is separated based on a preset row separation criterion, and the row separation criterion is expressed as:

1) T0=T′0∪T′1∪... ∪T′n−1

2) ∩T′j={ } for all 0≦≠j<n

where Ti denotes a sequence indicating locations of which permutation matrixes are arranged on the parent parity check matrix, and T′i denotes a sequence indicating a location on which a permutation matrix is arranged on a related row among n rows if the Ti is separated into the n rows.

13. The method as claimed in claim 8, wherein the row separation operation is an operation in which each of the rows included in the parent parity check matrix is separated based on a preset row separation criterion, if the row separation criterion indicates each of all rows included in the parent parity check matrix is separated into n rows, a jth row is generated by separating Ti={(ti,0,ei,0), (ti,1,ei,1),..., (Di−1,Di−1)} is mapped to an (nxi+j)th row included in a parity check matrix generated by performing the row separation operation on the parent parity check matrix, and the (nxi+j)th row included in a parity check matrix is expressed as:

x+j={()|0≦k<Di,k mod n=(n−1−j)},

where Ti denotes a sequence indicating locations of which permutation matrixes are arranged on the parent parity check matrix, T′i denotes a sequence indicating location on which a permutation matrix is arranged on a related row among n rows if the Ti is separated into the n rows, and ti,Di−1 denotes a location of a permutation matrix included in an ith row among the permutation matrixes included in the parent parity check matrix, and ei,Di−1 denotes an exponent of a permutation matrix located at a location indicated by ti,Di−1.

14. The signal transmission apparatus of claim 13, wherein the parent parity check matrix is generated using a base matrix,

wherein if a number of rows included in the base matrix is 2 and a number of columns included in the base matrix is 10, the parent parity check matrix is expressed as: T0={(0,0),(1,7),(2,4),(3,1),(4,6)} T1={(5,3),(6,0),(7,1),(8,6),(9,7)}, and

an 8×8 permutation matrix is substituted for each element included in the base matrix, the base matrix is expressed as: R0={0,1,2,3,4} R1={5,6,7,8,9},

where Ri denotes a sequence indicating locations at which elements have a ‘1’ value on the base matrix, ri,Di−1 as elements included in the Ri denotes an element included in the Ri and an index of a column with a value 1 on an ith row, and the Di denotes a degree of the ith row.

15. A signal transmission apparatus in a multimedia communication system, comprising:

a quasi-cyclic Low Density Parity Check (LDPC) code generator for generating a quasi-cyclic LDPC code; and

a transmitter for transmitting the quasi-cyclic LDPC code to a signal reception apparatus,

wherein the quasi-cyclic LDPC code is generated by encoding an information word vector using a child parity check matrix which is generated by performing one of a scaling operation, a row separation operation and a row merge operation on a parent parity check matrix, and

wherein the scaling operation is an operation in which a size of the child parity check matrix is determined, the row separation operation is an operation in which each of rows included in the parent parity check matrix is separated, and the row merge operation is an operation in which the rows included in the parent parity check matrix are merged.

16. The signal transmission apparatus as claimed in claim 15, wherein the parent parity check matrix includes an information part mapping to the information word vector and is generated using a base matrix.

17. The signal transmission apparatus as claimed in claim 15, wherein the row separation operation is an operation in which each of the rows included in the parent parity check matrix is separated based on a preset row separation criterion, the parent parity check matrix is generated using a base matrix, and the row separation criterion is determined using information related to permutation matrixes included in the parent parity check matrix and information related to a specific permutation matrix among permutation matrixes included in the base matrix.

18. The signal transmission apparatus as claimed in claim 17, wherein the information related to the permutation matrixes included in the parent parity check matrix includes information on a sequence indicating locations of which the permutation matrixes included in the parent parity check matrix are arranged on the parent parity check matrix, and the information related to the specific permutation matrix among permutation matrixes included in the base matrix includes information on an exponent of a permutation matrix arranged in a location at which a specific row among rows included in the base matrix and a specific column among columns included in the base matrix cross.

19. The signal transmission apparatus as claimed in claim 15, wherein the row separation operation is an operation in which each of the rows included in the parent parity check matrix is separated based on a preset row separation criterion, and the row separation criterion is expressed as:

1) T0=T′0∪T′1∪... ∪T′n−1

2) ∩T′j={ } for all 0≦≠j<n

where T1 denotes a sequence indicating locations of which permutation matrixes are arranged on the parent parity check matrix, and T′l denotes a sequence indicating a location on which a permutation matrix is arranged on a related row among n rows if the Ti is separated into the n rows.

20. The signal transmission apparatus as claimed in claim 15, wherein the row separation operation is an operation in which each of the rows included in the parent parity check matrix is separated based on a preset row separation criterion, if the row separation criterion indicates each of all rows included in the parent parity check matrix is separated into n rows, a jth row is generated by separating Ti={(ti,0,ei,0), (ti,1,ei,1),..., (Di−1,Di−1)} is mapped to an (nxi+j)th row included in a parity check matrix generated by performing the row separation operation on the parent parity check matrix, and the (nxi+j)th row included in a parity check matrix is expressed as:

x+j={()|0≦k<Di,k mod n=(n−1−j)},

where Ti denotes a sequence indicating locations of which permutation matrixes are arranged on the parent parity check matrix, T′i denotes a sequence indicating location on which a permutation matrix is arranged on a related row among n rows if the Ti is separated into the n rows, and ti,Di−1 denotes a location of a permutation matrix included in an ith row among the permutation matrixes included in the parent parity check matrix, and ei,Di−1 denotes an exponent of a permutation matrix located at a location indicated by ti,Di−1.

21. The signal transmission apparatus as claimed in claim 20, wherein the parent parity check matrix is generated using a base matrix,

wherein if a number of rows included in the base matrix is 2 and a number of columns included in the base matrix is 10, the parent parity check matrix is expressed as: T0={(0,0),(1,7),(2,4),(3,1),(4,6)} T1={(5,3),(6,0),(7,1),(8,6),(9,7)}, and

an 8×8 permutation matrix is substituted for each element included in the base matrix, the base matrix is expressed as: R0={0,1,2,3,4} R1={5,6,7,8,9},

where Ri denotes a sequence indicating locations at which elements have a ‘1’ value on the base matrix, ri,Di−1 as elements included in the Ri denotes an element included in the Ri and an index of a column with a value 1 on an ith row, and the Di denotes a degree of the ith row.

22. A signal reception apparatus in a multimedia communication system, comprising:

a receiver for receiving a quasi-cyclic Low Density Parity Check (LDPC) code; and

a LDPC code decoder for recovering an information word vector by decoding the quasi-cyclic LDPC code,

wherein the quasi-cyclic LDPC code is generated by encoding an information word vector using a child parity check matrix which is generated by performing one of a scaling operation, a row separation operation and a row merge operation on a parent parity check matrix, and

wherein the scaling operation is an operation in which a size of the child parity check matrix is determined, the row separation operation is an operation in which each of rows included in the parent parity check matrix is separated, and the row merge operation is an operation in which the rows included in the parent parity check matrix are merged.

23. The signal reception apparatus as claimed in claim 22, wherein the parent parity check matrix includes an information part mapping to the information word vector and is generated using a base matrix.

24. The signal reception apparatus as claimed in claim 22, wherein the row separation operation is an operation in which each of the rows included in the parent parity check matrix is separated based on a preset row separation criterion, the parent parity check matrix is generated using a base matrix, and the row separation criterion is determined using information related to permutation matrixes included in the parent parity check matrix and information related to a specific permutation matrix among permutation matrixes included in the base matrix.

25. The signal reception apparatus as claimed in claim 24, wherein the information related to the permutation matrixes included in the parent parity check matrix includes information on a sequence indicating locations of which the permutation matrixes included in the parent parity check matrix are arranged on the parent parity check matrix, and the information related to the specific permutation matrix among permutation matrixes included in the base matrix includes information on an exponent of a permutation matrix arranged in a location at which a specific row among rows included in the base matrix and a specific column among columns included in the base matrix cross.

26. The signal reception apparatus as claimed in claim 22, wherein the row separation operation is an operation in which each of the rows included in the parent parity check matrix is separated based on a preset row separation criterion, and the row separation criterion is expressed as:

1) T0=T′0∪T′1∪... ∪T′n−1

2) ∩T′j={ } for all 0≦≠j<n

where Ti denotes a sequence indicating locations of which permutation matrixes are arranged on the parent parity check matrix, and T′i denotes a sequence indicating a location on which a permutation matrix is arranged on a related row among n rows if the Ti is separated into the n rows.

27. The signal reception apparatus as claimed in claim 22, wherein the row separation operation is an operation in which each of the rows included in the parent parity check matrix is separated based on a preset row separation criterion, if the row separation criterion indicates each of all rows included in the parent parity check matrix is separated into n rows, a jth row is generated by separating Ti={(ti,0,ei,0), (ti,1,ei,1),..., (Di−1,Di−1)} is mapped to an (nxi+j)th row included in a parity check matrix generated by performing the row separation operation on the parent parity check matrix, and the (nxi+j)th row included in a parity check matrix is expressed as:

x+j={()|0≦k<Di,k mod n=(n−1−j)},

where Ti denotes a sequence indicating locations of which permutation matrixes are arranged on the parent parity check matrix, T′i denotes a sequence indicating location on which a permutation matrix is arranged on a related row among n rows if the Ti is separated into the n rows, and ti,Di−1 denotes a location of a permutation matrix included in an ith row among the permutation matrixes included in the parent parity check matrix, and ei,Di−1 denotes an exponent of a permutation matrix located at a location indicated by ti,Di−1.

28. The signal reception apparatus as claimed in claim 27, wherein the parent parity check matrix is generated using a base matrix,

wherein if a number of rows included in the base matrix is 2 and a number of columns included in the base matrix is 10, the parent parity check matrix is expressed as: T0={(0,0),(1,7),(2,4),(3,1),(4,6)} T1={(5,3),(6,0),(7,1),(8,6),(9,7)}, and

an 8×8 permutation matrix is substituted for each element included in the base matrix, the base matrix is expressed as: R0={0,1,2,3,4} R1={5,6,7,8,9},

where Ri denotes a sequence indicating locations at which elements have a ‘1’ value on the base matrix, ri,Di−1 as elements included in the Ri denotes an element included in the Ri and an index of a column with a value 1 on an ith row, and the Di denotes a degree of the ith row.